Question

Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u (x,0)-0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following ODE where G() is the Fourier transform of g(x) and U(w, t) is the Fourier transform of u(x, t) b) Solve the above ODE for U(a, t). Show that the solution is given by G(w) 2 1 Where C is a constant of integration? HINT: To solve the above equation, rewrite it as dU (w,t) - -(w2 + 1)dt and integrate on both sides to achieve the solution c) Use the initial condition to solve for the constant C. d) Assuming, g(x) e-x, we use entry 5 in Fourier transform tables to find that G()VTe-"*/4 Use this to show that the solution for U(w,t) can be rewritten as

Answer 1

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