2. a. Solve the diffusion equation with convection: J²u Ju = k- Ju +c Ət მე2 მე x>x>x- u(x, 0) = f(x). Hint: Apply the Fourier transform and then use the convolution theorem and shift theorem presented in part (b) of the previous question. b. Sketch the solution u(x, t) found in part (a) of the previous question for t = 1, t = 2, and t = 3 given that c = 2, k = 1, and the initial condition f(x)=6(x), where 8(x) is the Dirac delta function. ди Also, make a brief comment on the effect of the convection term, c, on the Əx' solution. Hint: Use the same idea found in the eText (page 452) when considering the Dirac delta function (x) to find the fundamental solution of the heat equation.
2. a. Solve the diffusion equation with convection: J²u Ju = k- Ju +c Ət მე2 მე x>x>x- u(x, 0) = f(x). Hint: Apply the Fourier transform and then use the convolution theorem and shift theorem presented in part (b) of the previous question. b. Sketch the solution u(x, t) found in part (a) of the previous question for t = 1, t = 2, and t = 3 given that c = 2, k = 1, and the initial condition f(x)=6(x), where 8(x) is the Dirac delta function. ди Also, make a brief comment on the effect of the convection term, c, on the Əx' solution. Hint: Use the same idea found in the eText (page 452) when considering the Dirac delta function (x) to find the fundamental solution of the heat equation.
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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Please solve the following by hand and without the use of AI, Please use detailed mathematical expression to solve and explain each step as you go. Ensure the answer is correct please. Thank you!
Transcribed Image Text:2. a. Solve the diffusion equation with convection:
J²u
Ju
= k-
Ju
+c
Ət
მე2
მე
x>x>x-
u(x, 0) = f(x).
Hint: Apply the Fourier transform and then use the convolution theorem and
shift theorem presented in part (b) of the previous question.
b. Sketch the solution u(x, t) found in part (a) of the previous question for
t = 1, t = 2, and t = 3 given that c = 2, k = 1, and the initial condition
f(x)=6(x), where 8(x) is the Dirac delta function.
ди
Also, make a brief comment on the effect of the convection term, c, on the
Əx'
solution.
Hint: Use the same idea found in the eText (page 452) when considering the
Dirac delta function (x) to find the fundamental solution of the heat
equation.
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