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Rouben Rostamian

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19 years, 3 days
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These are answers submitted by Rouben Rostamian

Solution in sine series...

Rouben Rostamian 7711
July 29 2019
3 1
> 

restart;
> 

de_orig := r*diff(H(r),r,r) + diff(H(r),r) + (k^2*r - b^2/R^2*r^2)*H(r);

r*(diff(diff(H(r), r), r))+diff(H(r), r)+(k^2*r-b^2*r^2/R^2)*H(r)

> 

bc := H(R)=0, H(1/R)=R;

H(R) = 0, H(1/R) = R

We change the independent variable from r to rho so that domain R, 1/R is mapped to "(0,Pi)."
Additionally, we change the dependent variable from H to G so that G(0) = G((π) = 0.

> 

change_of_vars := { r = (R^2 - 1)*rho/(Pi*R) + 1/R,
                    H(r) = G(rho) + R/Pi*(Pi-rho) };

{r = (R^2-1)*rho/(Pi*R)+1/R, H(r) = G(rho)+R*(Pi-rho)/Pi}

> 

tmp1 := PDEtools:-dchange(change_of_vars, [de_orig,bc], [G(rho), rho]);

[((R^2-1)*rho/(Pi*R)+1/R)*(diff(diff(G(rho), rho), rho))*Pi^2*R^2/(R^2-1)^2+(diff(G(rho), rho)-R/Pi)*Pi*R/(R^2-1)+(k^2*((R^2-1)*rho/(Pi*R)+1/R)-b^2*((R^2-1)*rho/(Pi*R)+1/R)^2/R^2)*(G(rho)+R*(Pi-rho)/Pi), G(Pi) = 0, G(0)+R = R]

The new differential equation is:

> 

de := tmp1[1];

((R^2-1)*rho/(Pi*R)+1/R)*(diff(diff(G(rho), rho), rho))*Pi^2*R^2/(R^2-1)^2+(diff(G(rho), rho)-R/Pi)*Pi*R/(R^2-1)+(k^2*((R^2-1)*rho/(Pi*R)+1/R)-b^2*((R^2-1)*rho/(Pi*R)+1/R)^2/R^2)*(G(rho)+R*(Pi-rho)/Pi)

Let's verify that the boundary conditions are G(0) = G(Pi) and G(Pi) = 0, as expected:

> 

tmp1[2];

G(Pi) = 0

> 

tmp1[3];

G(0)+R = R

Now that the domain is 0, Pi and the boundary conditions are zero, we may
express the solution in an infinite sine series:

> 

S := sum(a[n]*sin(n*rho), n=1..infinity);

sum(a[n]*sin(n*rho), n = 1 .. infinity)

To determine the coefficients a__n, plug the series into the differential equation:

> 

eval(de, G(rho)=S):
tmp2 := combine(%);

(sum((-sin(n*rho)*Pi^4*R^7*n^2*rho*a[n]+sin(n*rho)*Pi^2*R^9*k^2*rho*a[n]-sin(n*rho)*Pi*R^8*b^2*rho^2*a[n]+cos(n*rho)*Pi^4*R^7*n*a[n]-sin(n*rho)*Pi^5*R^5*n^2*a[n]+sin(n*rho)*Pi^4*R^5*n^2*rho*a[n]+sin(n*rho)*Pi^3*R^7*k^2*a[n]-3*sin(n*rho)*Pi^2*R^7*k^2*rho*a[n]-2*sin(n*rho)*Pi^2*R^6*b^2*rho*a[n]+4*sin(n*rho)*Pi*R^6*b^2*rho^2*a[n]-Pi^4*R^5*a[n]*n*cos(n*rho)-2*sin(n*rho)*Pi^3*R^5*k^2*a[n]+3*Pi^2*R^5*k^2*rho*a[n]*sin(n*rho)-sin(n*rho)*Pi^3*R^4*b^2*a[n]+6*sin(n*rho)*Pi^2*R^4*b^2*rho*a[n]-6*rho^2*R^4*b^2*Pi*a[n]*sin(n*rho)+Pi^3*R^3*k^2*a[n]*sin(n*rho)-R^3*k^2*rho*Pi^2*a[n]*sin(n*rho)+2*sin(n*rho)*Pi^3*R^2*b^2*a[n]-6*R^2*b^2*rho*Pi^2*a[n]*sin(n*rho)+4*rho^2*R^2*b^2*Pi*a[n]*sin(n*rho)-b^2*Pi^3*a[n]*sin(n*rho)+2*b^2*Pi^2*a[n]*sin(n*rho)*rho-b^2*Pi*a[n]*sin(n*rho)*rho^2)/(R^2-1), n = 1 .. infinity)-Pi*R^8*k^2*rho^2-Pi*R^7*b^2*rho^2-3*Pi^2*R^6*k^2*rho+2*Pi*R^6*k^2*rho^2-2*Pi^2*R^5*b^2*rho+5*Pi*R^5*b^2*rho^2+2*Pi^2*R^4*k^2*rho-Pi*R^4*k^2*rho^2+5*Pi^2*R^3*b^2*rho-7*Pi*R^3*b^2*rho^2-3*Pi^2*R*b^2*rho+3*Pi*R*b^2*rho^2+Pi^2*R^8*k^2*rho-3*R^5*b^2*rho^3+3*R^3*b^2*rho^3-R*b^2*rho^3-Pi^3*R^4*k^2-Pi^3*R^3*b^2+R^7*b^2*rho^3+Pi^3*R^6*k^2-Pi^3*R^6+Pi^3*R*b^2)/(Pi^3*R^6-Pi^3*R^4)

We want to determine the coefficients a__n so that tmp2 is identically zero on 0 < rho and rho < Pi.
For that purpose, we multiple tmp2 by sin(m*rho) and integrate the result from 0 to Pi.
That results in equations of the form
`&alpha;__m`+`&beta;__m`*a__m+sum(`&gamma;__m,n`*a__n, n = 1 .. infinity) = 0,    m = 1, 2, () .. ()
where the constants `&alpha;__m`, `&beta;__m`, `&gamma;__m,n` can be computed explicitly.  This is a coupled linear system
of infinitely many equations in the infinitely many unknowns a__m.  We cannot afford solving
that system but we can do the next best thing, which is to replace infinity with a finite number N 
and solve the resulting linear system of N equations in N unknowns.  That way you may obtain
a solution to any desired degree of accuracy.

The calculation is not difficult but it is tedious, so I leave it to you to finish it.

 


Download de.mw

Solution through finite differences...

Rouben Rostamian 7711
July 28 2019
3 1

Maple 2019 does not have a numeric solver for elliptic PDEs.
That does not prevent us from writing our own finite-difference
code.  Here is an illustration of solving Laplace's equation in
polar coordinates.

> 

restart;
> 

with(LinearAlgebra):
> 

with(plots):

The domain is a < r and r < b, c < t and t < d in polar coordinates.
For illustration we take:

> 

a := 1:  b :=2:
c := Pi/6:  d := Pi/2:

We subdivide the r interval into n__r subintervals of equal lengths.
We subdivide the tinterval into `n__t `subintervals of equal lengths.

> 

n__r := 10;
n__t := 15;

10

15

Calculate the resulting mesh sizes in the r and t directions:

> 

delta__r := (b-a)/n__r;
delta__t := (d-c)/n__t;

1/10

(1/45)*Pi

Here is what the mesh looks like:

> 

seq(seq([(a+i*delta__r)*cos(c+j*delta__t), (a+i*delta__r)*sin(c+j*delta__t)],
    i=0..n__r), j=0..n__t):
pointplot([%], symbol=solidcircle, color="Green", scaling=constrained);

We wish to solve the PDE

> 

pde := diff(u(r,t),r,r) + diff(u(r,t),r)/r + diff(u(r,t),t,t)/r^2 =  0;

diff(diff(u(r, t), r), r)+(diff(u(r, t), r))/r+(diff(diff(u(r, t), t), t))/r^2 = 0

on that domain, subject to the Dirichlet boundary conditions
"u(r,c)=`alpha__1`(r), u(r,d)=`alpha__2`(r), a<r<b,"
u(a, t) = `&alpha;__3`(t), u(b, t) = `&alpha;__4`(t), c < t and t < d.

We discretize the PDE in the interior points through central differencing.
Here U[i, j] represents the solution at the point of index i, j

> 

discretized_pde :=
   (U[i-1,j] - 2*U[i,j] + U[i+1,j])/delta__r^2
 + 1/(a+i*delta__r)*(U[i+1,j] - U[i-1,j])/(2*delta__r)
 + 1/(a+i*delta__r)^2*(U[i,j-1] - 2*U[i,j] + U[i,j+1])/delta__t^2 = 0;

100*U[i-1, j]-200*U[i, j]+100*U[i+1, j]+5*(U[i+1, j]-U[i-1, j])/((1/10)*i+1)+2025*(U[i, j-1]-2*U[i, j]+U[i, j+1])/(((1/10)*i+1)^2*Pi^2) = 0

That discretization results in (n__r-1)*(n__t-1) equations in the unknowns U[i, j]:

> 

eq_interior := seq(seq(discretized_pde, i=1..n__r-1), j=1..n__t-1):

Additionally, we have 2*n__t+2*n__r equations that assign boundary values to U[i, j] 

> 

eq_boundary :=
  seq(U[i,0]=1,                    i=1..n__r-1),        # u(r,c)=1
  seq(U[i,n__t]=0,                 i=1..n__r-1),        # u(r,d)=0
  seq(U[0,j]=0,                    j=0..n__t),          # u(a,t)=0
  seq(U[n__r,j]=c+j*delta__t,      j=0..n__t):          # u(b,t)=t

Altogether, these provide a system of (n__r+1)*(n__t+1)linear equations
in the (n__r+1)*(n__t+1) unknowns U[i, j]

> 

sys := eq_interior, eq_boundary:
> 

vars := seq(seq(U[i,j], j=0..n__t), i=0..n__r):
> 

nops([sys]), nops([vars]);

176, 176

Extract the system's coefficient matrix:

> 

LinSys := GenerateMatrix(evalf([sys]), [vars], augmented,
    datatype=float[8], storage=sparse):

Solve the system:

> 

V := LinearSolve(LinSys):

The vector Vcalculated above is of length (n__r+1)*(n__t+1).  We reformat
its entries into an `&x`(n__r+1, n__t+1)table T:

> 

Matrix(n__r+1, n__t+1, convert(V, list), datatype=float[8]):
T := rtable_redim(%, 0..n__r, 0..n__t):

Finally, we plot the solution:

> 

dataplot(a..b, c..d, T, coords=cylindrical,
    scaling=constrained, orientation=[-72,50,0]);


 

Download finite-difference-elliptic-polar.mw

typesetprime...

Rouben Rostamian 7711
July 27 2019
4 0
> 

restart;
> 

kernelopts(version);

`Maple 2019.1, X86 64 LINUX, May 21 2019, Build ID 1399874`

> 

Typesetting:-Settings(typesetprime=true):
> 

diff(y(x),x);

diff(y(x), x)

> 

diff(y(x),x,x);

diff(diff(y(x), x), x)

 

Perhaps this?...

Rouben Rostamian 7711
July 27 2019
1 1 
restart;
with(plots):
frame := proc(i)
    local t := 22 - i;
    plot(-i*x/t+i, x=0..20, y=0..20, color=blue, thickness=3);
end proc:
display(seq(display(seq(frame(j), j=0..i)), i=0..20), insequence);

Perhaps this will do...

Rouben Rostamian 7711
July 25 2019
1 0 
A := 35.438*y1(x) - 34424*y2(x) + 2742.0234*z1(x) + 78.34;
remove(has, A, x);

 

Solution...

Rouben Rostamian 7711
July 17 2019
1 0

As Carl has pointed out, answers to your questions are available in the explicit solution which was given in your earlier question.  Nevertheless, here are the details.

Your statement about x__1 is correct, but the ratio of  the two x__2 solutions is not infinity.

 

Here is your system of differential equations:

> 

restart;
> 

[-p[1]*x[1]^2+x[2], -2*p[1]^2*x[1]^3+2*p[1]*x[1]*x[2]+x[1]+1]:
f := unapply(%, [x[1],x[2]]):
diff~([x[1](t), x[2](t)], t) =~ f(x[1](t), x[2](t)):
sys := %[];

diff(x[1](t), t) = -x[1](t)^2*p[1]+x[2](t), diff(x[2](t), t) = -2*x[1](t)^3*p[1]^2+2*x[1](t)*x[2](t)*p[1]+x[1](t)+1

and the initial conditions

> 

ic := x[1](0) = p[2], x[2](0) = p[3];

x[1](0) = p[2], x[2](0) = p[3]

Solve the system:

> 

dsolve({sys, ic}, {x[1](t), x[2](t)}):
dsol := convert(%, trigh);

{x[1](t) = (p[2]+1)*cosh(t)-1+(-p[1]*p[2]^2+p[3])*sinh(t), x[2](t) = (p[2]+1)^2*p[1]*cosh(t)^2+(-2*(p[1]*p[2]^2-p[3])*(p[2]+1)*p[1]*sinh(t)-p[2]^2*p[1]-2*p[1]*p[2]-2*p[1]+p[3])*cosh(t)+(p[1]*p[2]^2-p[3])^2*p[1]*sinh(t)^2+(2*p[1]^2*p[2]^2-2*p[1]*p[3]+p[2]+1)*sinh(t)+p[1]}

Extract the components x__1 and x__2:

> 

x1_p := eval(x[1](t), dsol);
x2_p := eval(x[2](t), dsol);

(p[2]+1)*cosh(t)-1+(-p[1]*p[2]^2+p[3])*sinh(t)

(p[2]+1)^2*p[1]*cosh(t)^2+(-2*(p[1]*p[2]^2-p[3])*(p[2]+1)*p[1]*sinh(t)-p[2]^2*p[1]-2*p[1]*p[2]-2*p[1]+p[3])*cosh(t)+(p[1]*p[2]^2-p[3])^2*p[1]*sinh(t)^2+(2*p[1]^2*p[2]^2-2*p[1]*p[3]+p[2]+1)*sinh(t)+p[1]

If we solve the system with a different set of parameters, say[q__1, q__2, q__3], the solution will be

> 

x1_q := eval(x1_p, p=q);
x2_q := eval(x2_p, p=q);

(q[2]+1)*cosh(t)-1+(-q[1]*q[2]^2+q[3])*sinh(t)

(q[2]+1)^2*q[1]*cosh(t)^2+(-2*(q[1]*q[2]^2-q[3])*(q[2]+1)*q[1]*sinh(t)-q[1]*q[2]^2-2*q[1]*q[2]-2*q[1]+q[3])*cosh(t)+(q[1]*q[2]^2-q[3])^2*q[1]*sinh(t)^2+(2*q[1]^2*q[2]^2-2*q[1]*q[3]+q[2]+1)*sinh(t)+q[1]

You want x1_p and x1_q to be the same, that is:

> 

x1_p = x1_q;

(p[2]+1)*cosh(t)-1+(-p[1]*p[2]^2+p[3])*sinh(t) = (q[2]+1)*cosh(t)-1+(-q[1]*q[2]^2+q[3])*sinh(t)

So the coefficients of cosh(t) on the two sides should be the same.  Also those of sinh(t). That is:

> 

tmp1 :=
p[2]+1 = q[2]+1,
-p[1]*p[2]^2+p[3] = -q[1]*q[2]^2+q[3];

p[2]+1 = q[2]+1, -p[1]*p[2]^2+p[3] = -q[1]*q[2]^2+q[3]

Solve for q__2 and q__3:

> 

tmp2 := solve({tmp1}, {q[2], q[3]});

{q[2] = p[2], q[3] = -p[1]*p[2]^2+p[2]^2*q[1]+p[3]}

That agrees with what you have written.

We evaluate the solution x2_q with this choice of the parameters
and then look at the ratio x2_q / x2_p as t goes to infinity:

> 

R := eval(x2_q, tmp2) / x2_p:
> 

limit(R, t=infinity);

q[1]/p[1]

We see that the limit is finite.


 

Download mw.mw

A solution via differential equations...

Rouben Rostamian 7711
July 16 2019
2 3

As an alternative to mmcdara's solution, here is an approach
via differential equations.
Let's write g(x) for the integrand and let f(a) = int(g(x), x = 0 .. a).
You wish to determine a so that the integral evaluates to 0.5.
From the definition of f we have diff(f(x), x) = g(x), f(0) = 0. We may
view this as an initial value problem, and use dsolve()'s  events
option to stop the integration when the integral reaches 0.5.

> 

restart;
> 

g := x -> sqrt(1+(-5.557990765*sin(5.557990765*x)-7.3*cos(5.557990765*x)-5.6*sinh(5.557990765*x)+7.3*cosh(5.557990765*x))^2):
> 

de := diff(f(x),x) = g(x);

diff(f(x), x) = (1+(-5.557990765*sin(5.557990765*x)-7.3*cos(5.557990765*x)-5.6*sinh(5.557990765*x)+7.3*cosh(5.557990765*x))^2)^(1/2)

> 

dsol := dsolve({de,f(0)=0}, numeric, events=[[f(x)=0.5, halt]]);

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 26, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([Array(1..2, 1..21, {(1, 1) = 1.0, (1, 2) = .0, (1, 3) = 1.0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (1, 7) = 1.0, (1, 8) = undefined, (1, 9) = -.5, (1, 10) = 1.0, (1, 11) = undefined, (1, 12) = undefined, (1, 13) = undefined, (1, 14) = undefined, (1, 15) = undefined, (1, 16) = undefined, (1, 17) = undefined, (1, 18) = undefined, (1, 19) = undefined, (1, 20) = undefined, (1, 21) = undefined, (2, 1) = 1.0, (2, 2) = .0, (2, 3) = 100.0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (2, 7) = .0, (2, 8) = undefined, (2, 9) = undefined, (2, 10) = 0.10e-6, (2, 11) = undefined, (2, 12) = .0, (2, 13) = undefined, (2, 14) = .0, (2, 15) = .0, (2, 16) = undefined, (2, 17) = undefined, (2, 18) = undefined, (2, 19) = undefined, (2, 20) = undefined, (2, 21) = undefined}, datatype = float[8], order = C_order), proc (x, Y, Ypre, n, EA) EA[1, 7+2*n] := Y[1]-.5; EA[1, 8+2*n] := 1; 0 end proc, proc (e, x, Y, Ypre) return 0 end proc, Array(1..1, 1..2, {(1, 1) = undefined, (1, 2) = undefined}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..63, {(1) = 1, (2) = 1, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 1, (17) = 0, (18) = 1, (19) = 30000, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = 0.5047658755841546e-2, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..1, {(1) = .0}, datatype = float[8], order = C_order)), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..1, {(1) = .1}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 0}, datatype = integer[8]), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 1.0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..1, {(1, 1) = .0, (2, 0) = .0, (2, 1) = .0, (3, 0) = .0, (3, 1) = .0, (4, 0) = .0, (4, 1) = .0, (5, 0) = .0, (5, 1) = .0, (6, 0) = .0, (6, 1) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = f(x)]`; YP[1] := evalf((1+(-5.557990765*sin(5.557990765*X)-7.3*cos(5.557990765*X)-5.6*sinh(5.557990765*X)+7.3*cosh(5.557990765*X))^2)^(1/2)); 0 end proc, -1, 0, 0, 0, 0, proc (x, Y, Ypre, n, EA) EA[1, 7+2*n] := Y[1]-.5; EA[1, 8+2*n] := 1; 0 end proc, proc (e, x, Y, Ypre) return 0 end proc, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = f(x)]`; YP[1] := evalf((1+(-5.557990765*sin(5.557990765*X)-7.3*cos(5.557990765*X)-5.6*sinh(5.557990765*X)+7.3*cosh(5.557990765*X))^2)^(1/2)); 0 end proc, -1, 0, 0, 0, 0, proc (x, Y, Ypre, n, EA) EA[1, 7+2*n] := Y[1]-.5; EA[1, 8+2*n] := 1; 0 end proc, proc (e, x, Y, Ypre) return 0 end proc, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0) ] )) ] ); _y0 := Array(0..1, {(1) = 0.}); _vmap := array( 1 .. 1, [( 1 ) = (1) ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); _i := false; if _par <> [] then _i := `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then _i := `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) or _i end if; if _i then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch: end try else try procname(procname("right")) catch: end try end if end if end if; return elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch: end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch: end try end if end if end if; return elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error end try; if _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [x, f(x)], (4) = []}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol := 1; _ndsol := _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error end try end proc

> 

plots:-odeplot(dsol, x=0..1):

Warning, cannot evaluate the solution further right of .15500879, event #1 triggered a halt

So at x = .155 .. () we hit the target.  Let's verify that indeed the integral reaches 0.5
at that point:

> 

evalf(Int(g(x), x=0 .. 0.15500879));

.4999999462


 

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Answer...

Rouben Rostamian 7711
July 15 2019
1 1

This question has been answered before.  See:

https://www.mapleprimes.com/questions/227376-Imaginary-To-Real

ideas...

Rouben Rostamian 7711
July 13 2019
1 1

I can tell you how things work under Linux.  They should not be very different under Windows.

Maple's system() and ssystem() functions pass their arguments as commands to the underlying operating system for execution. System() returns the command's exit status (usually 0 for success), while ssystem() returns to command's output as a Maple string. If I understand you correctly, you are interested in system(), or rather more specifically, in system[launch](), where the launch option spawns a new process, and therefore Maple does not wait for the command's termination.

For instance, to open a new terminal and set its title to "My title", I would type on the Linux commandline (not in Maple) the command:

xterm -title "My title"

To do the same from within Maple, I would do:

system[launch]("xterm", "-title", "My title");

The first argument (xterm) is the command to be executed.  The remaining arguments are options passed to that command.

Things should work similarly under Windows but note that system()'s help page says that under Windows "The given command should specify an application or executable file, not a shell or DOS command". See what you can do.

 

Solution...

Rouben Rostamian 7711
July 13 2019
3 0

Your conjecture is almost correct, but not quite.

 

I will paraphrase your statement and show how to analyze it.

 

You have a system of differential equations:

> 

restart;
> 

[-p[1]*x[1]^2+x[2], -2*p[1]^2*x[1]^3+2*p[1]*x[1]*x[2]+x[1]+1]:
f := unapply(%, [x[1],x[2]]):
diff~([x[1](t), x[2](t)], t) =~ f(x[1](t), x[2](t)):
sys := %[];

diff(x[1](t), t) = -x[1](t)^2*p[1]+x[2](t), diff(x[2](t), t) = -2*x[1](t)^3*p[1]^2+2*x[1](t)*x[2](t)*p[1]+x[1](t)+1

and initial conditions

> 

ic := x[1](0) = p[2], x[2](0) = p[3];

x[1](0) = p[2], x[2](0) = p[3]

The system may be solved symbolically.  We don't need numerics:

> 

dsolve({sys, ic}, {x[1](t), x[2](t)}):
dsol := convert(%, trigh);

{x[1](t) = (p[2]+1)*cosh(t)-1+(-p[1]*p[2]^2+p[3])*sinh(t), x[2](t) = (p[2]+1)^2*p[1]*cosh(t)^2+(-2*(p[1]*p[2]^2-p[3])*(p[2]+1)*p[1]*sinh(t)-p[2]^2*p[1]-2*p[1]*p[2]-2*p[1]+p[3])*cosh(t)+(p[1]*p[2]^2-p[3])^2*p[1]*sinh(t)^2+(2*p[1]^2*p[2]^2-2*p[1]*p[3]+p[2]+1)*sinh(t)+p[1]}

You are interested in the following special choice of parameters,
where k is a positive integer:

> 

params := p[1]=k, p[2]=2, p[3]=4*k-1;

p[1] = k, p[2] = 2, p[3] = 4*k-1

so we evaluate the solution with those parameters:

> 

mydsol := eval(dsol, [p[1]=k, p[2]=2, p[3]=4*k-1]);

{x[1](t) = 3*cosh(t)-1-sinh(t), x[2](t) = 9*k*cosh(t)^2+(-6*k*sinh(t)-6*k-1)*cosh(t)+k*sinh(t)^2+(8*k^2-2*k*(4*k-1)+3)*sinh(t)+k}

 

As you have conjectured correctly, x1(t) is independent of k.  That's good!

You have also conjectured that

x__2(t)*`|at any k` = k*x__2(t)*`|at k=1`

That conjecture is true asymptotically for large values of t but it
breaks down for small t .

To see that, we calculate:

> 

B := eval(x[2](t), mydsol):  # x[2](t) at any k
A := eval(B, k=1):           # x[2](t) at k=1
R := unapply(B/A, k);        # their ratio
 

 

proc (k) options operator, arrow; (9*k*cosh(t)^2+(-6*k*sinh(t)-6*k-1)*cosh(t)+k*sinh(t)^2+(8*k^2-2*k*(4*k-1)+3)*sinh(t)+k)/(9*cosh(t)^2+(-6*sinh(t)-7)*cosh(t)+sinh(t)^2+5*sinh(t)+1) end proc

For large t the ratio approaches k, so the conjecture is almost true in that sense:

> 

limit(R(k), t=infinity);

k

but for small k the ratio is different fromk -- it oscillates a little before it approaches k

> 

plot([seq(R(k), k=1..5)], t=0..8, view=0..7);


 

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Increase computational acuracy...

Rouben Rostamian 7711
July 12 2019
1 1

Increase computational accuracy by specifying a larger value for Digits, as in:
 

Digits := 20;

The the result will be 9.96087e+16.both in GUI and terminal.

Try this...

Rouben Rostamian 7711
July 12 2019
1 0

In the absence of further information, this is the best that can be done.  A good answer can be very different depending on how you want to use the result.

# returns 0 if b is even, else 1
if_odd := b -> `if`(type(b,even),0,1):
# answer
ans := 4*theta[j]/N*Sum(Sum('if_odd(n+l)'*n*theta[n]/c[l]*T[n](x[j])*T[l](x[k]),
   l=0..n-1), n=0..N);

 

Alternative version...

Rouben Rostamian 7711
July 09 2019
2 5

This is pretty much like Kitonum's but just a little more compact:

Eq := v1(t) - R1*(-i3(t)/n13 - i2(t)/n12) - L1*diff(-i3(t)/n13 - i2(t)/n12, t) = n12*(v2(t) - R2*i2(t) - L2*diff(i2(t), t));
expand(lhs(Eq) - rhs(Eq)):
select(has, %, diff) = -remove(has, %, diff):
collect(%, diff);

 

No solution...

Rouben Rostamian 7711
July 07 2019
3 0

Plot EQ:

plot(EQ, x=0.9e8..1.1e8, discont=true);

We see that the graph jumps from a negative value to a positive value at 1e8.  There is no intersection with the x axis, therefore no root.

Depending on your application, there may be reason to take 1e8 as a root, but that justification, if there is one, is external to mathematics.

 

Solution...

Rouben Rostamian 7711
July 06 2019
4 3

The functions Q(x, z, t) and W(x, z, t) satisfy your PDEs.  Let's introduce the
auxiliary functions phi(x, z, t) and psi(x, z, t) as the laplacians of Q and W:
"phi(x,z,t)=((&PartialD;)^(2 )W)/((&PartialD;)^( )x^2)+((&PartialD;)^(2 )W)/((&PartialD;)^( )z^2)",
"psi(x,z,t)=((&PartialD;)^(2 )Q)/((&PartialD;)^( )x^2)+((&PartialD;)^(2 )Q)/((&PartialD;)^( )z^2)".
I will show that phi and psi satisfy the wave equations
"((&PartialD;)^2phi)/((&PartialD;)^( )t^2)=mu/(rho)(((&PartialD;)^2phi)/((&PartialD;)^( )x^2)+((&PartialD;)^2phi)/((&PartialD;)^( )z^2)),"
"((&PartialD;)^2psi)/((&PartialD;)^( )t^2)=(lambda+2 mu)/(rho)(((&PartialD;)^2psi)/((&PartialD;)^( )x^2)+((&PartialD;)^2psi)/((&PartialD;)^( )z^2)),"
Here is how.

> 

restart;
> 

pde1 := rho*(diff(diff(Q(x,z,t),t,t),x)) + rho*(diff(diff(W(x,z,t),t,t),z))
      - (lambda+2*mu)*(diff(diff(Q(x,z,t),x,x) + diff(Q(x,z,t),z,z),x))
      -            mu*(diff(diff(W(x,z,t),x,x) + diff(W(x,z,t),z,z),z)) = 0;

rho*(diff(diff(diff(Q(x, z, t), t), t), x))+rho*(diff(diff(diff(W(x, z, t), t), t), z))-(lambda+2*mu)*(diff(diff(diff(Q(x, z, t), x), x), x)+diff(diff(diff(Q(x, z, t), x), z), z))-mu*(diff(diff(diff(W(x, z, t), x), x), z)+diff(diff(diff(W(x, z, t), z), z), z)) = 0

> 

pde2 := rho*(diff(diff(Q(x,z,t),t,t),z)) - rho*(diff(diff(W(x,z,t),t,t), x))
      - (lambda+2*mu)*(diff(diff(Q(x,z,t),x,x) + diff(Q(x,z,t),z,z),z))
      +            mu*(diff(diff(W(x,z,t),x,x) + diff(W(x,z,t),z,z),x)) = 0;

rho*(diff(diff(diff(Q(x, z, t), t), t), z))-rho*(diff(diff(diff(W(x, z, t), t), t), x))-(lambda+2*mu)*(diff(diff(diff(Q(x, z, t), x), x), z)+diff(diff(diff(Q(x, z, t), z), z), z))+mu*(diff(diff(diff(W(x, z, t), x), x), x)+diff(diff(diff(W(x, z, t), x), z), z)) = 0

Following Edgardo's tip we apply casesplit() with the optional argument [Q,W]:

> 

casesplit1 := PDEtools:-casesplit([pde1,pde2], [Q,W]);

`casesplit/ans`([diff(diff(diff(Q(x, z, t), x), x), x) = (-lambda*(diff(diff(diff(Q(x, z, t), x), z), z))-2*mu*(diff(diff(diff(Q(x, z, t), x), z), z))-mu*(diff(diff(diff(W(x, z, t), x), x), z))-mu*(diff(diff(diff(W(x, z, t), z), z), z))+rho*(diff(diff(diff(Q(x, z, t), t), t), x))+rho*(diff(diff(diff(W(x, z, t), t), t), z)))/(lambda+2*mu), diff(diff(diff(Q(x, z, t), x), x), z) = (rho*(diff(diff(diff(Q(x, z, t), t), t), z))-rho*(diff(diff(diff(W(x, z, t), t), t), x))-lambda*(diff(diff(diff(Q(x, z, t), z), z), z))-2*mu*(diff(diff(diff(Q(x, z, t), z), z), z))+mu*(diff(diff(diff(W(x, z, t), x), x), x))+mu*(diff(diff(diff(W(x, z, t), x), z), z)))/(lambda+2*mu), diff(diff(diff(diff(W(x, z, t), x), x), x), x) = (rho*(diff(diff(diff(diff(W(x, z, t), t), t), x), x))-2*mu*(diff(diff(diff(diff(W(x, z, t), x), x), z), z))-mu*(diff(diff(diff(diff(W(x, z, t), z), z), z), z))+rho*(diff(diff(diff(diff(W(x, z, t), t), t), z), z)))/mu], [])

Pick the last equation returned by casesplit:

> 

op([1,-1],casesplit1):
rhs(%) - lhs(%):
Eq1_tmp := expand(%*mu/rho);

diff(diff(diff(diff(W(x, z, t), t), t), x), x)-2*mu*(diff(diff(diff(diff(W(x, z, t), x), x), z), z))/rho-mu*(diff(diff(diff(diff(W(x, z, t), z), z), z), z))/rho+diff(diff(diff(diff(W(x, z, t), t), t), z), z)-mu*(diff(diff(diff(diff(W(x, z, t), x), x), x), x))/rho

In terms of the function "phi, "the above leads to the following wave equation with the wave speed sqrt(mu/rho):

> 

Eq1 := diff(phi(x,z,t), t,t) = mu/rho * (diff(phi(x,z,t),x,x) +diff(phi(x,z,t),z,z));

diff(diff(phi(x, z, t), t), t) = mu*(diff(diff(phi(x, z, t), x), x)+diff(diff(phi(x, z, t), z), z))/rho

Let's verify that Eq1_tmp and Eq1 are equivalent:

> 

lhs(Eq1)-rhs(Eq1) - Eq1_tmp:
simplify(%, {phi(x,z,t)=diff(W(x,z,t),x,x) + diff(W(x,z,t),z,z)});

0

OK, they are.  Now apply casesplit() once again, but this time with the reversed optional argument:

> 

casesplit2 := PDEtools:-casesplit([pde1,pde2], [W,Q]);

`casesplit/ans`([diff(diff(diff(W(x, z, t), x), x), x) = (-rho*(diff(diff(diff(Q(x, z, t), t), t), z))+rho*(diff(diff(diff(W(x, z, t), t), t), x))+lambda*(diff(diff(diff(Q(x, z, t), x), x), z))+2*mu*(diff(diff(diff(Q(x, z, t), x), x), z))+lambda*(diff(diff(diff(Q(x, z, t), z), z), z))+2*mu*(diff(diff(diff(Q(x, z, t), z), z), z))-mu*(diff(diff(diff(W(x, z, t), x), z), z)))/mu, diff(diff(diff(W(x, z, t), x), x), z) = (rho*(diff(diff(diff(W(x, z, t), t), t), z))+rho*(diff(diff(diff(Q(x, z, t), t), t), x))-mu*(diff(diff(diff(W(x, z, t), z), z), z))-lambda*(diff(diff(diff(Q(x, z, t), x), x), x))-2*mu*(diff(diff(diff(Q(x, z, t), x), x), x))-lambda*(diff(diff(diff(Q(x, z, t), x), z), z))-2*mu*(diff(diff(diff(Q(x, z, t), x), z), z)))/mu, diff(diff(diff(diff(Q(x, z, t), x), x), x), x) = (rho*(diff(diff(diff(diff(Q(x, z, t), t), t), x), x))-2*lambda*(diff(diff(diff(diff(Q(x, z, t), x), x), z), z))-4*mu*(diff(diff(diff(diff(Q(x, z, t), x), x), z), z))-lambda*(diff(diff(diff(diff(Q(x, z, t), z), z), z), z))-2*mu*(diff(diff(diff(diff(Q(x, z, t), z), z), z), z))+rho*(diff(diff(diff(diff(Q(x, z, t), t), t), z), z)))/(lambda+2*mu)], [])

Pick the last equation returned by casesplit:

> 

op([1,-1],casesplit2):
rhs(%) - lhs(%):
%*(lambda+2*mu):
Eq2_tmp := simplify(%, size);

(-lambda-2*mu)*(diff(diff(diff(diff(Q(x, z, t), x), x), x), x))+(-4*mu-2*lambda)*(diff(diff(diff(diff(Q(x, z, t), x), x), z), z))+(-lambda-2*mu)*(diff(diff(diff(diff(Q(x, z, t), z), z), z), z))+rho*(diff(diff(diff(diff(Q(x, z, t), t), t), x), x)+diff(diff(diff(diff(Q(x, z, t), t), t), z), z))

In terms of the function psi the above leads to the following wave equation with the wave speed sqrt((lambda+2*mu)/rho):

> 

Eq2 := diff(psi(x,z,t), t,t) = (lambda+2*mu)/rho * (diff(psi(x,z,t),x,x) +diff(psi(x,z,t),z,z));

diff(diff(psi(x, z, t), t), t) = (lambda+2*mu)*(diff(diff(psi(x, z, t), x), x)+diff(diff(psi(x, z, t), z), z))/rho

Let's verify that Eq2_tmp and Eq2 are equivalent:

> 

lhs(Eq2)-rhs(Eq2) - Eq2_tmp/rho:
simplify(%, {psi(x,z,t)=diff(Q(x,z,t),x,x) + diff(Q(x,z,t),z,z)});

0


 

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