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du d²u Consider the heat equation = at +2x, for a bar of length L=1, with non homogeneous boundary conditions: u(0,t) = 1 and u(1,t)=3. The initial 2x² condition is u(x,0)=x. The problem is transformed into another one with homogeneous conditions, in terms of V=U-UE, where UE is the equilibrium solution of the u. Hence Əv 2²v and v(0,t) = 0 and v(1,t) = 0. Ət 2x² Hence, v(x, t) = Ž ansin(n πx) e-(π)²t n=1 Please write the solution on a paper, scan and upload. 1) Derive the equilibrium solution UE 2) Derive the initial condition v(x,0) 3) Evaluate the integral formula of an, and simplify the resulting expression 4) Calculate the the values of an, at n=1 and n=2 =

du d²u Consider the heat equation = at +2x, for a bar of length L=1, with non homogeneous boundary conditions: u(0,t) = 1 and u(1,t)=3. The initial 2x² condition is u(x,0)=x. The problem is transformed into another one with homogeneous conditions, in terms of V=U-UE, where UE is the equilibrium solution of the u. Hence Əv 2²v and v(0,t) = 0 and v(1,t) = 0. Ət 2x² Hence, v(x, t) = Ž ansin(n πx) e-(π)²t n=1 Please write the solution on a paper, scan and upload. 1) Derive the equilibrium solution UE 2) Derive the initial condition v(x,0) 3) Evaluate the integral formula of an, and simplify the resulting expression 4) Calculate the the values of an, at n=1 and n=2 =

Advanced Engineering Mathematics
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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du d'u Consider the heat equation = Ət +2x, for a bar of length L=1, with non homogeneous boundary conditions: u(0,t) = 1 and u(1,t) = 3. The initial 2x² condition is u(x,0)=x. The problem is transformed into another one with homogeneous conditions, in terms of v=u-UE, where UE is the equilibrium solution of the u. Hence Əv 2²v = and v(0,t) = 0 and v(1,t) = 0. dt Əx² Hence, v(x,t) = Σ a sin(nèx) e−(nπ)²t n=1 Please write the solution on a paper, scan and upload. 1) Derive the equilibrium solution UE 2) Derive the initial condition v(x,0) 3) Evaluate the integral formula of an, and simplify the resulting expression 4) Calculate the the values of an, at n=1 and n=2
Transcribed Image Text:du d'u Consider the heat equation = Ət +2x, for a bar of length L=1, with non homogeneous boundary conditions: u(0,t) = 1 and u(1,t) = 3. The initial 2x² condition is u(x,0)=x. The problem is transformed into another one with homogeneous conditions, in terms of v=u-UE, where UE is the equilibrium solution of the u. Hence Əv 2²v = and v(0,t) = 0 and v(1,t) = 0. dt Əx² Hence, v(x,t) = Σ a sin(nèx) e−(nπ)²t n=1 Please write the solution on a paper, scan and upload. 1) Derive the equilibrium solution UE 2) Derive the initial condition v(x,0) 3) Evaluate the integral formula of an, and simplify the resulting expression 4) Calculate the the values of an, at n=1 and n=2
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