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Dkunb

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These are questions asked by Dkunb

Integration of Hermite to Laguerre...

Dkunb 50 Maple 2022
integration hermite
February 27 2024
1 3

How  to make the integration 

 

into the following form:



Thanks in advance,Q1.mw

How to find the normalization factor ?...

Dkunb 50 Maple
solve
February 17 2024
2 0

SolnforOS_dsolve.mw 
I would like to solve eq6 in terms of sigma and n. I would appreciate for your help.

How to find the solutions matrix(array)...

Dkunb 50 Maple 2021
array ode syntax
June 13 2023
1 11

How to find Z(t) matrix(array) =[ [z[1,0](t),z[1,1](t)],[z[2,0](t),z[2,1](t)]]

> 

restart;
> 

with(IntegrationTools):with(Physics):
> 

 

> 

 

> 

 

> 

h1 := (m,n)->(n+1/2)*KroneckerDelta[n,m];

proc (m, n) options operator, arrow; Physics:-`*`(n+Physics:-`*`(1, Physics:-`^`(2, -1)), Physics:-KroneckerDelta[n, m]) end proc

 (1)
> 

h2 := (mu,nu,m2,l)->-(nu*Pi/l)^2/(2*m2)*KroneckerDelta[mu,nu];

proc (mu, nu, m2, l) options operator, arrow; Physics:-`*`(-1, Physics:-`*`(Physics:-`^`(Physics:-`*`(nu, Pi, Physics:-`^`(l, -1)), 2), Physics:-`^`(Physics:-`*`(2, m2), -1), Physics:-KroneckerDelta[mu, nu])) end proc

 (2)
> 

v1 := (m,n,m1)->sqrt(min(n,m)!/max(n,m)!)*(2*m1)^(-abs(n-m)/2)*exp(-1/(4*m1))*LaguerreL(min(n,m),abs(n-m),-1/(2*m1));

proc (m, n, m1) options operator, arrow; Physics:-`*`(sqrt(Physics:-`*`(factorial(min(n, m)), Physics:-`^`(factorial(max(n, m)), -1))), Physics:-`^`(Physics:-`*`(2, m1), Physics:-`*`(-1, Physics:-`*`(abs(n-m), Physics:-`^`(2, -1)))), exp(Physics:-`*`(-1, Physics:-`*`(1, Physics:-`^`(Physics:-`*`(4, m1), -1)))), LaguerreL(min(n, m), abs(n-m), Physics:-`*`(-1, Physics:-`*`(1, Physics:-`^`(Physics:-`*`(2, m1), -1))))) end proc

 (3)
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v2 := (mu,nu,l)->4*Pi^2*l*mu*nu*(exp(l/2)-(-1)^(mu+nu)*exp(-l/2))/((Pi*(mu+nu))^2+l^2)/((Pi*(mu-nu))^2+l^2);

proc (mu, nu, l) options operator, arrow; Physics:-`*`(4, Physics:-`^`(Pi, 2), l, mu, nu, exp(Physics:-`*`(l, Physics:-`^`(2, -1)))-Physics:-`*`(Physics:-`^`(-1, mu+nu), exp(Physics:-`*`(-1, Physics:-`*`(l, Physics:-`^`(2, -1))))), Physics:-`^`(Physics:-`^`(Physics:-`*`(Pi, mu+nu), 2)+Physics:-`^`(l, 2), -1), Physics:-`^`(Physics:-`^`(Physics:-`*`(Pi, mu-nu), 2)+Physics:-`^`(l, 2), -1)) end proc

 (4)
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h:=(m,n,mu,nu,m1,m2,l)->evalf(h1(m,n)+h2(mu,nu,m2,l)+v1(m,n,m1)+v2(mu,nu,l));

proc (m, n, mu, nu, m1, m2, l) options operator, arrow; evalf(h1(m, n)+h2(mu, nu, m2, l)+v1(m, n, m1)+v2(mu, nu, l)) end proc

 (5)
> 

m1:=1:m2:=1:l:=1:
> 

H:= (m,n,mu,nu)->h(m,n,mu,nu,m1,m2,l);

proc (m, n, mu, nu) options operator, arrow; h(m, n, mu, nu, m1, m2, l) end proc

 (6)
> 

H(1,1,0,0);

2.668201175

 (7)
> 

eq1:= diff(z(m,nu,t),t)=-I*Sum(Sum(H(m,n,mu,nu)*z(n,mu,t),n=1..N),mu=0..M);

diff(z(m, nu, t), t) = -I*(Sum(Sum(((n+.5000000000)*Physics:-KroneckerDelta[m, n]-4.934802202*nu^2*Physics:-KroneckerDelta[mu, nu]+.7788007831*(factorial(min(m, n))/factorial(max(m, n)))^(1/2)*2.^(-.5000000000*abs(-1.*n+m))*LaguerreL(min(m, n), abs(-1.*n+m), -.5000000000)+39.47841762*mu*nu*(1.648721271-.6065306597*(-1.)^(mu+nu))/((9.869604404*(mu+nu)^2+1.)*(9.869604404*(mu-1.*nu)^2+1.)))*z(n, mu, t), n = 1 .. N), mu = 0 .. M))

 (8)
> 

 

> 

zint := Array([[1,0],[0,1]]);

Matrix(2, 2, {(1, 1) = 1, (1, 2) = 0, (2, 1) = 0, (2, 2) = 1})

 (9)
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Z:= Array(1..2,1..2);
 for i from 1to 2 do
    for j from 1 to 2 do
        Z[i,j]:= dsolve({eq1,zint[i,j]},numeric,output=listprocedure);
    end do;
end do;
> 

 

Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 0})

  

Error, (in dsolve/numeric/process_input) system must be entered as a set/list of expressions/equations

  
> 

 

Download test3.mw

better way to write the code for the int...

Dkunb 50 Maple 2021
integration orthogonal
June 12 2023
1 1

how to write the code for integration of the orthogonal cosine function with nu=mu or with nu<>mu

> 

restart;
 
> 

phi:= (mu,Q2)->sqrt(2/l)*sin(mu*Pi*(Q2+l/2)/l);    # E:=mu->(Pi*mu/sqrt(2)/l)^2, mu=1,2...

proc (mu, Q2) options operator, arrow; sqrt(2/l)*sin(mu*Pi*(Q2+(1/2)*l)/l) end proc

 (1)
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fh1:=simplify((-1/2/m2*Int(diff(phi(mu,Q2),Q2)*diff(phi(nu,Q2),Q2),Q2=-l/2..l/2)))

-mu*Pi^2*nu*(Int(cos((1/2)*mu*Pi*(2*Q2+l)/l)*cos((1/2)*nu*Pi*(2*Q2+l)/l), Q2 = -(1/2)*l .. (1/2)*l))/(l^3*m2)

 (2)
> 

convert(fh1,int) assuming(mu,integer,nu,integer);

0

 (3)
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fh1_subs := simplify(subs(nu = mu, fh1));

-mu^2*Pi^2*(Int(cos((1/2)*mu*Pi*(2*Q2+l)/l)^2, Q2 = -(1/2)*l .. (1/2)*l))/(l^3*m2)

 (4)
> 

convert(fh1_substituted,int)assuming (mu,integer,nu,integer);

-(1/2)*mu^2*Pi^2/(l^2*m2)

 (5)
> 

 

Download test1.mw

ode system is singular at the initial po...

Dkunb 50 Maple 2021
ode dsolve numeric
April 13 2023
2 1

Could you help me how to deal with this problem?

> 

restart;
> 

with(plots):with(plottools):with(DETools):
> 

 

> 

Sys:=diff(T(R),R)=((1-1/R)*(sqrt(1-(alpha/R)^2*(1-1/R))))^(-1),diff(Phi(R),R)=(alpha/R)^2*(sqrt(1-(alpha/R)^2*(1-1/R)))^(-1);

diff(T(R), R) = 1/((1-1/R)*(1-alpha^2*(1-1/R)/R^2)^(1/2)), diff(Phi(R), R) = alpha^2/(R^2*(1-alpha^2*(1-1/R)/R^2)^(1/2))

 (1)
> 

 

> 

inits:=[[T(0)=0.5,Phi(0)=0],[T(0)=0.5,Phi(0)=Pi/4]];

[[T(0) = .5, Phi(0) = 0], [T(0) = .5, Phi(0) = (1/4)*Pi]]

 (2)
> 

K:=dsolve([Sys,op(op(1,inits))],[Phi(R),T(R)],numeric,parameters=[alpha],output=listprocedure);

Error, (in dsolve/numeric/make_proc) ode system is singular at the initial point

  
> 

 

Download HW6.m

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