• Introduction
• Operators
• Solution
• Example

Green functions for regular differential operators

by Aleksas Domarkas

Vilnius Lniversity, Faculty of Mathematics and Informatics,

Naugarduko 24, Vilnius, Lithuania

aleksas@ieva.mif.vu.lt

NOTE: In this session we find Green functions for regular differential

operators Au=-diff(p(x)*diff(u(x),x),x)+q(x)*u(x).

Introduction

>

restart;

Please  input  number of examples n (1..12)  and   Execute Worksheet

>

n:=3;

Operators

>	L||1:=-diff(u(x),x,x),u(0)=0,u(1)=0;

>	L||2:=-diff(u(x),x,x),u(0)=0,D(u)(1)=0;

>	L||3:=-diff(u(x),x,x),u(0)=0,u(1)+beta*D(u)(1)=0;

>	L||4:=-diff(u(x),x,x)-u(x),u(0)=0,u(1)=0;

>	L||5:=-diff(u(x),x,x)-u(x),u(0)-D(u)(0)=0,u(1)-D(u)(1)=0;

>	L||6:=-diff(u(x),x,x)+u(x),u(0)=0,u(1)=0;

>	L||7:=-diff(u(x),x,x)+u(x),D(u)(0)=0,D(u)(1)=0;

>	L||8:=-(1+x^2)*diff(u(x),x,x)-2*x*diff(u(x),x),u(0)-D(u)(0)=0,u(1)=0;

>	L||9:=-(1+x^2)*diff(u(x),x,x)-2*x*diff(u(x),x),u(0)=0,u(1)+D(u)(1)=0;

>	L||10:=-Diff(Diff(u(x),x)/(x-2),x)+3*u(x)/(x-2)^3,u(0)=0,u(1)=0;

>	L||11:=-Diff(Diff(u(x),x)*exp(-x^2/2),x)+exp(-x^2/2)*u(x),u(0)=0,u(1)=0;

>	L||12:=-diff(u(x),x,x)+(1+x^2)*u(x),u(0)=0,D(u)(1)=0;

>

Solution

>	l:=value(L||n[1]);p(x):=-coeff(l,diff(u(x),x,x));

>	s1:=L||n[2]; s2:=L||n[3];

>	rhs(dsolve(l,u(x)));

>	u:=unapply(%,x);

>

s1;

>	simplify(solve(%,{_C1,_C2}));

>	y1:=unapply(simplify(subs(%,u(x))),x);

>

s2;

>	simplify(solve(%,{_C1,_C2}));

>	y2:=unapply(simplify(subs(%,u(x))),x);

>	w(x) :=simplify( y1(x)*diff(y2(x),x)-y2(x)*diff(y1(x),x));

>	k:=simplify(subs(x=0,p(x)*w(x)));

>	G:=(x,y)->piecewise(x<=y, -y1(x)*(y2(y))/k, x>=y, -y1(y)*(y2(x))/k);

>

G(x,y);

>	G(x,y):=normal(%);u:='u':

Example

>	cat(`EXAMPLE `, n); L||n;

>	`G(x,y)`=G(x,y);

>	#plot3d(G(x,y), x=0..1, y=0..1);

>

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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