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2. a. Solve the diffusion equation with convection: {?u/?t = k ?²u/?x² + c ?u/?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) = ?(x), where ?(x) is the Dirac delta function. Also, make a brief comment on the effect of the convection term, c ?u/?x, on the solution. Hint: Use the same idea found in the textbook (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:

{?u/?t = k ?²u/?x² + c ?u/?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) = ?(x), where ?(x) is the Dirac delta function.

Also, make a brief comment on the effect of the convection term, c ?u/?x, on the solution.

Hint: Use the same idea found in the textbook (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:

?u/?t = k ?²u/?x² + c ?u/?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) = ?(x), where ?(x) is the Dirac delta function.

Also, make a brief comment on the effect of the convection term, c ?u/?x, on the solution.

Hint: Use the same idea found in the textbook (page 452) when considering the Dirac delta function ?(x) to find the fundamental solution of the heat equation.

Added by Charles T.

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