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radsymell.mws

Radial-symmetric solutions of linear elliptic equations

by Aleksas Domarkas

Vilnius University, Faculty of Mathematics and Informatics,

Naugarduko 24, Vilnius, Lithuania

aleksas@ieva.mif.vu.lt

NOTE: In this session we find radial-symmetric solutions of linear elliptic equations.

Introduction

>    restart;

Please  input  number of examples k (1..6),  dimension  n (2, 3, ...) and   Execute Worksheet

>    k:=2;  n:=3;

k := 2

n := 3

Operators

Examples of elliptic  operators:

>    L[1]:=sum(D[i]@@2,i=1..n):#Laplacian;

>    L[2]:=sum(D[i]@@2,i=1..n)+(x->4*x):#Helmholtz equation

>    L[3]:=L[1]@L[1]:#Laplacian^2;

>    L[4]:=L[1]@L[2]:

>    L[5]:=L[3]-L[1]:

>    L[6]:=L[1]@@3:#Laplacian^3

>   

Solution

>    L:=L[k];

L := `@@`(D[1],2)+`@@`(D[2],2)+`@@`(D[3],2)+proc (x) options operator, arrow; 4*x end proc

>    x := ('x||i' $ i =1..n);

x := x1, x2, x3

>    F:=unapply(u(x),x);

F := u

>    X2 := sum( 'x||i^2', i=1..n);

X2 := x1^2+x2^2+x3^2

>    f := unapply( g(X2), x);

f := proc (x1, x2, x3) options operator, arrow; g(x1^2+x2^2+x3^2) end proc

>    Lu:=convert(expand(L(F)(x)),diff):

>    w := simplify( L(f)(x), {X2=t} );

w := 4*`@@`(D,2)(g)(t)*t+6*D(g)(t)+4*g(t)

>    dsolve( w, g(t) );

g(t) = _C1/t^(1/2)*sin(2*sqrt(t))+_C2/t^(1/2)*cos(2*sqrt(t))

>    f:=map(simplify,rhs(%));

f := _C1/t^(1/2)*sin(2*sqrt(t))+_C2/t^(1/2)*cos(2*sqrt(t))

Example

The radial-symmetric solution of equation

>    Lu=0;

diff(u(x1,x2,x3),`$`(x1,2))+diff(u(x1,x2,x3),`$`(x2,2))+diff(u(x1,x2,x3),`$`(x3,2))+4*u(x1,x2,x3) = 0

is

>    u=f;

u = _C1/t^(1/2)*sin(2*sqrt(t))+_C2/t^(1/2)*cos(2*sqrt(t))

where

>    t=X2;

t = x1^2+x2^2+x3^2

>   

Checking the Solution

>    Lu=0;

diff(u(x1,x2,x3),`$`(x1,2))+diff(u(x1,x2,x3),`$`(x2,2))+diff(u(x1,x2,x3),`$`(x3,2))+4*u(x1,x2,x3) = 0

>    sol:=u(x)=subs(t=X2,f);

sol := u(x1,x2,x3) = _C1/(x1^2+x2^2+x3^2)^(1/2)*sin(2*sqrt(x1^2+x2^2+x3^2))+_C2/(x1^2+x2^2+x3^2)^(1/2)*cos(2*sqrt(x1^2+x2^2+x3^2))

>    simplify(subs(sol,Lu));

0

>   

While every effort has been made to validate the solutions in this worksheet, Waterloo Maple Inc. and the contributors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material.

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