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Solve Uxx + Uyy = f(x, y), ux (0, y) = F(y), u(x,0) = f(x), 0

Solve Uxx + Uyy = f(x, y), ux (0, y) = F(y), u(x,0) = f(x), 0

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 15EQ
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Problem 1: Solve Uxr + Uyy = f (x, y), 0 < x < a, 0 < x < b, Uz (0, y) = F(y), u(a, y) = G(y) u(x, 0) = f(x), u(x, b) = g(x)
Transcribed Image Text:Problem 1: Solve Uxr + Uyy = f (x, y), 0 < x < a, 0 < x < b, Uz (0, y) = F(y), u(a, y) = G(y) u(x, 0) = f(x), u(x, b) = g(x)
Problem 2: In steady-state heat transfer (or in electrostatics), Laplace equation, V²u = 0, is valid when the medium is homogeneous (i.e., the medium is independent on spatial variables). For a nonhomogeneous medium, Laplace equation generalizes to V²u + • Vu = 0, where v(x, y, z) is a function describing the medium properties. For a homogeneous medium, v = c, where c is a constant. Consider a rectangle filled with a medium described by v(x) = e" and excited from one side by the source f(x). Then, the problem can be described by Uxx + Ug + Uyy 0, 0 < x < a, 0 < x < b, u(0, y) = 0, u (а, у) — = 0 u(x, 0) = f(x), u(x, b) = 0 Solve it.
Transcribed Image Text:Problem 2: In steady-state heat transfer (or in electrostatics), Laplace equation, V²u = 0, is valid when the medium is homogeneous (i.e., the medium is independent on spatial variables). For a nonhomogeneous medium, Laplace equation generalizes to V²u + • Vu = 0, where v(x, y, z) is a function describing the medium properties. For a homogeneous medium, v = c, where c is a constant. Consider a rectangle filled with a medium described by v(x) = e" and excited from one side by the source f(x). Then, the problem can be described by Uxx + Ug + Uyy 0, 0 < x < a, 0 < x < b, u(0, y) = 0, u (а, у) — = 0 u(x, 0) = f(x), u(x, b) = 0 Solve it.
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