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Mariusz Iwaniuk

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9 years, 262 days

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These are answers submitted by Mariusz Iwaniuk

I have only:...

Mariusz Iwaniuk 1476
March 15 2024
1 0


 

> 

restart
> 

kernelopts(version)

`Maple 2024.0, X86 64 WINDOWS, Mar 01 2024, Build ID 1794891`

 (1)
> 

F := int(1/(u*sqrt(1+p1*u^2/(2*p2))), u)

-arctanh(2^(1/2)/(2+p1*u^2/p2)^(1/2))

 (2)
> 

G := `assuming`([simplify(F)], [p1 > 0, p2 < 0])

arctanh(2^(1/2)*p2/((p1*u^2+2*p2)*p2)^(1/2))

 (3)
> 

S := solve(G = x-x[0], u)

(-2*p1*p2*(tanh(x-x[0])^2-1))^(1/2)/(p1*tanh(x-x[0])), -(-2*p1*p2*(tanh(x-x[0])^2-1))^(1/2)/(p1*tanh(x-x[0]))

 (4)
> 

`assuming`([simplify({evalc(Im(S[1])), evalc(Im(S[2]))})], [p1 > 0, p2 < 0, x-x[0] > 0])

{2^(1/2)*(-p2)^(1/2)*csch(x-x[0])/p1^(1/2), -2^(1/2)*(-p2)^(1/2)*csch(x-x[0])/p1^(1/2)}

 (5)
> 

NULL


 

Download start_v1.mw

Maple is not figure it out to give solut...

Mariusz Iwaniuk 1476
February 28 2024
1 5
> 

restart;
> 

eq1 := 2^(-m/2-n/2)*exp(lambda^2*sigma^2/4)/sqrt(n!)/sqrt(m!*Pi)*int(HermiteH(m,s+lambda*sigma/2)*HermiteH(n,s+lambda*sigma/2)*exp(-s^2), s=-infinity..infinity);;

2^(-(1/2)*m-(1/2)*n)*exp((1/4)*lambda^2*sigma^2)*(int(HermiteH(m, s+(1/2)*lambda*sigma)*HermiteH(n, s+(1/2)*lambda*sigma)*exp(-s^2), s = -infinity .. infinity))/(factorial(n)^(1/2)*(factorial(m)*Pi)^(1/2))

 (1)
> 

eq2 := 2^(-(1/2)*n+(1/2)*m)*exp((1/4)*lambda^2*sigma^2)*sqrt(factorial(m))*(lambda*sigma)^(-m+n)*LaguerreL(m, -m+n, -(1/2)*lambda^2*sigma^2)/sqrt(factorial(n))
> 

eq1 = eq2

2^(-(1/2)*m-(1/2)*n)*exp((1/4)*lambda^2*sigma^2)*(int(HermiteH(m, s+(1/2)*lambda*sigma)*HermiteH(n, s+(1/2)*lambda*sigma)*exp(-s^2), s = -infinity .. infinity))/(factorial(n)^(1/2)*(factorial(m)*Pi)^(1/2)) = 2^(-(1/2)*n+(1/2)*m)*exp((1/4)*lambda^2*sigma^2)*factorial(m)^(1/2)*(lambda*sigma)^(-m+n)*LaguerreL(m, -m+n, -(1/2)*lambda^2*sigma^2)/factorial(n)^(1/2)

 (2)
> 

NULL
> 

NULL
> 

m := 2; n := 5; lambda := 1/6; sigma := 2/3
> 

evalf[20](eq1)

0.62998679107780885597e-3

 (3)
> 

evalf[20](eq2)

0.62998679107780885597e-3

 (4)
> 

``

Download Q2.mw

Only simple equation...

Mariusz Iwaniuk 1476
February 22 2024
1 0

We can use Laplace transform and Inverse Laplace transfrom to solve  for: Simple linear-differential  fractional-equations with initial conditions.

differential_equations_with_fractional_order.mw

Solve delay differential system of equat...

Mariusz Iwaniuk 1476
January 30 2024
0 2

Maple can solve only numerically.Adding random missing values to parameters:

dsolve-delay_sys_example_1.mw

I use Maple version 2023.2,I don't have version 18 !

Regards

For n=10000...

Mariusz Iwaniuk 1476
October 21 2023
1 0
> 

restart;
> 

ee := unapply((-1)^n*((-4*n^2 - 16*n - 28)*JacobiP(-1 + n, -1 - 2*n, 2, -1/2) + JacobiP(-2 + n, -2*n, 3, -1/2)*(3 + n)*(-1 + n))*4^n/(48*(1 + n)*n),n):
> 

eee := rsolve({a(1) = 1, a(2) = 2, a(n) = ((10*n-16)*a(n-1)-(9*n-27)*a(n-2))/(n-1)}, a, 'makeproc')
> 

L:=seq(evalb(expand(ee(i))=eee(i+1)), i=1..10000):
> 

ListTools:-Occurrences(true, [L])

10000

 (1)

Download hg_ex.mw

After 30 min computation on my hardware we see that for n =10000 are True.

Using Laplace transform...

Mariusz Iwaniuk 1476
September 04 2023
4 3
> 

restart
> 

ElzakiTransform := proc (f, t) simplify(inttrans:-laplace(f*v, t, 1/v)) end proc; f := exp(n*t); g := ElzakiTransform(f, t)

v/(1/v-n)

 (1)
> 

InverseElzakiTransform := proc (g, v) inttrans:-invlaplace(eval(g/v, v = 1/v), v, t) end proc
> 

`assuming`([InverseElzakiTransform(g, v)], [n > 0])

exp(n*t)

 (2)
> 

NULL
> 

restart
> 

ElzakiTransform := proc (f, t) simplify(inttrans:-laplace(f*v, t, 1/v)) end proc; f := exp(-t^2); g := ElzakiTransform(f, t)

-(1/2)*v*Pi^(1/2)*exp((1/4)/v^2)*(-1+erf((1/2)/v))

 (3)
> 

InverseElzakiTransform := proc (g, v) inttrans:-invlaplace(eval(g/v, v = 1/v), v, t) end proc
> 

InverseElzakiTransform(g, v)

exp(-t^2)

 (4)
> 

> 

restart
> 

ElzakiTransform := proc (f, t) simplify(inttrans:-laplace(f*v, t, 1/v)) end proc; f := sin(t); g := ElzakiTransform(f, t)

v^3/(v^2+1)

 (5)
> 

InverseElzakiTransform := proc (g, v) inttrans:-invlaplace(eval(g/v, v = 1/v), v, t) end proc
> 

InverseElzakiTransform(g, v)

sin(t)

 (6)
> 

> 

restart
> 

ElzakiTransform := proc (f, t) simplify(inttrans:-laplace(f*v, t, 1/v)) end proc; f := 1/(1+t); g := ElzakiTransform(f, t)

v*exp(1/v)*Ei(1, 1/v)

 (7)
> 

InverseElzakiTransform := proc (g, v) inttrans:-invlaplace(eval(g/v, v = 1/v), v, t) end proc
> 

InverseElzakiTransform(g, v)

1/(1+t)

 (8)
> 

NULL

Download Elzaki_Transfrom.mw

Try....

Mariusz Iwaniuk 1476
August 29 2023
0 0

 

kernelopts(version);

#`Maple 2023.1, X86 64 WINDOWS, Jul 07 2023, Build ID 1723669`

Try:

allvalues(value(solll));

# long answer

If want solve for: y(x) :

solve(simplify(allvalues(value(solll))), [y(x)]);

#very long answer

 

Syntax.....

Mariusz Iwaniuk 1476
August 17 2023
1 0

Try:

with(DynamicSystems):
lambda := 15.4;
f := N1 -> sum(exp(-lambda)*lambda^n/n!, n = 1 .. N1);
Ns := 50;
T := Vector(Ns, t -> t);
A := Vector(Ns, t -> f(t));
DiscretePlot(T, A, style = stair, legend = "stair", color = red, labels = ["time", "signal"]);
plot(f(x), x = 0 .. Ns);

.

with(DynamicSystems);
f := N1 -> sum(1/(n^3*sin(n)^2), n = 1 .. N1);
Ns := 400;
T := Vector(Ns, t -> t);
A := Vector(Ns, t -> f(t));
DiscretePlot(T, A, style = stair, legend = "stair", color = red, labels = ["time", "signal"]);

 

Maybe you what:...

Mariusz Iwaniuk 1476
August 11 2023
1 0

From help pages the fractional derivative using the Davison-Essex (D-E) definition:

diff(f(x),[x$nu]) = 1/GAMMA(n-nu)*Int((x-t)^(n-nu-1)*diff(f(t),[t$n]),t = 0 .. x);

 

U1 := t -> (1/2*1/M - 1/4*1/(M*K))*t + 1/2;
eq := Int((t - z)^(ceil(alpha) - alpha - 1)*diff(U1(z), [z $ ceil(alpha)]), z = 0 .. t)/GAMMA(ceil(alpha) - alpha) + U1(t)/M - U1(t)^2/(M*K) + diff(U1(t), t) - diff(U1(t), t)/epsilon;
(value(eq) assuming (0 < alpha and alpha < 1));
int(%, t);

#t/(2*M) - t/(4*M*K) - ((2/M - 1/(M*K))*t)/(4*epsilon) + (2*M*K*t + t^2*K - 1/2*t^2)/(4*M^2*K) - (1 + (1/M - 1/(2*M*K))*t)^3/(12*M*K*(1/M - 1/(2*M*K))) - (2*K - 1)*t^(2 - alpha)/(4*(-1 + alpha)*M*K*GAMMA(1 - alpha)*(2 - alpha))

 

 

 

With Mathematica:...

Mariusz Iwaniuk 1476
June 22 2023
0 0

Where QPochhammer function.

FoxH function...

Mariusz Iwaniuk 1476
June 20 2023
2 1

Looks like analytical solution for sum is FoxH function:

sum(R^g*product(-B*r + N*g + 1, r = 1 .. g - 1)/(B^g*g!), g = 0 .. infinity) = -FoxH([[[1 + 1/B, 1 - N/B]], []], [[[0, 1]], [[1/B, -N/B]]], R)/B

See attached file.

brn_ac_ver2.mw

Analytical solution with Mathematica....

Mariusz Iwaniuk 1476
June 07 2023
0 0

See attached file:

1_case1.mw

Try:...

Mariusz Iwaniuk 1476
March 06 2023
1 3 
INV := invztrans((z - 1)^2/(a*z^2 + b*z + c), z, n);
simplify(allvalues(INV));

#(-2^(-n)*((a - c)*sqrt(-4*a*c + b^2) + (b + 4*c)*a + c*b)*((-b + sqrt(-4*a*c + b^2))/a)^n + ((-a + c)*sqrt(-4*a*c + b^2) + (b + 4*c)*a + c*b)*(-(b + sqrt(-4*a*c + b^2))/(2*a))^n + 2*charfcn[0](n)*a*sqrt(-4*a*c + b^2))/(2*sqrt(-4*a*c + b^2)*a*c)

 

Try:...

Mariusz Iwaniuk 1476
February 13 2023
2 0

 Because dsolve doesn't know whether to compute  T(x) or T(L ) .

dsolve({ics2, ode}, T(x));

 

Using Mathematica....

Mariusz Iwaniuk 1476
January 21 2023
2 5

If we use MMA to solve series:

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