4. Consider the following initial value problem of the 1D hcat equation with mixed boundary condition DE: u = kuz, 0 0, k>0, IC: u(r, t = 0) = g(x), 0 < r < l, ВС: и(0, t) — 0, и.(1,t) — 0, t>0. (a) Use the encrgy method to show that there is at most one solution for the initial-boundary value problem. (b) Suppose u(r, t) = X (x)T(t) is a seperable solution. Show that X and T satisfy X" = AX, T' = \kT, for some A E R. (c) Find all the cigenvalues A, and the corresponding eigenfunction X„() for the problem X" = AX with boundary condition X (0) = 0, X'(1) = 0. (d) Solve the initial-boundary value problem. Express the solution as series of the cigenfunctions {X„(x)}.

4. Consider the following initial value problem of the 1D hcat equation with mixed boundary condition DE: u = kuz, 0 0, k>0, IC: u(r, t = 0) = g(x), 0 < r < l, ВС: и(0, t) — 0, и.(1,t) — 0, t>0. (a) Use the encrgy method to show that there is at most one solution for the initial-boundary value problem. (b) Suppose u(r, t) = X (x)T(t) is a seperable solution. Show that X and T satisfy X" = AX, T' = \kT, for some A E R. (c) Find all the cigenvalues A, and the corresponding eigenfunction X„() for the problem X" = AX with boundary condition X (0) = 0, X'(1) = 0. (d) Solve the initial-boundary value problem. Express the solution as series of the cigenfunctions {X„(x)}.

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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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

4. Consider the following initial value problem of the 1D heat equation with mixed boundary
condition
kuga, 0<x <I, t>0, k > 0,
IC: u(x, t = 0) = 9(x), 0<x < I,
ВС: и(0, t) — 0, и.(1,t) — 0, t>0.
DE: ut =
(a) Use the energy method to show that there is at most one solution for the
initial-boundary value problem.
(b) Suppose u(x, t) = X (x)T(t) is a seperable solution. Show that X and T satisfy
X" = XX, T' = \kT,
for some A E R.
(c) Find all the cigenvalues A, and the corresponding eigenfunction X, (r) for the
problem X" = XX with boundary condition X (0) = 0, X'(1) = 0.
(d) Solve the initial-boundary value problem. Express the solution as series of the
eigenfunctions {X„(x)}.

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