Solve Completely to the final form of the 200Solution as a Fourier Series the heatequation boundary value problem. A(homogeneous PDĚ, Dirichlet, non-homogeneousbut constant B.C.)xe LOL] L: 4, kẸ 3 t 201R 2²u = 0B.C.IC2u242x² 10 xut foodgou(0₁t) = 5 $ roz sand on : You aliau(L₁Z) = -5AAu(x₂0) = x9wole9

Given Heat equation is the following :Let, so we get By substitution ,we get

Answered: Solve Completely to the final form of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

Solve Completely to the final form of the 200
Solution as a Fourier Series the heat
equation boundary value problem. A
(homogeneous PDĚ, Dirichlet, non-homogeneous
but constant B.C.)
xe LOL] L: 4, kẸ 3 t 201
R 2²u = 0
B.C.
IC
2u
24
2x² 10 xut food
go
u(0₁t) = 5 $ roz sand on : You alia
u(L₁Z) = -5
A
A
u(x₂0) = x
9
wole
9

Expert Solution

Step 1: Heat equation solve

Given Heat equation is the following :

$fraction numerator partial differential u over denominator partial differential t end fraction equals 3 fraction numerator partial differential squared u over denominator partial differential x squared end fraction comma 0 less or equal than x less or equal than 4 comma t greater than 0$

$u left parenthesis 0 comma t right parenthesis equals 5 comma u left parenthesis 4 comma t right parenthesis equals negative 5 comma t greater than 0 space a n d space$

$u left parenthesis x comma 0 right parenthesis equals x comma 0 less or equal than x less or equal than 4$

Let, $v left parenthesis x comma t right parenthesis equals u left parenthesis x comma t right parenthesis minus open curly brackets 5 plus x over 4 open parentheses negative 5 minus 5 close parentheses close curly brackets equals u left parenthesis x comma t right parenthesis plus 5 over 2 x minus 5$

so we get $v subscript t equals u subscript t space a n d space v subscript x x end subscript equals u subscript x x end subscript$

By substitution ,we get $v subscript t equals 3 v subscript x x end subscript$