Question

please provide me with full working solution. Any help is appreciated. thank you in advance

Consider the diffusion equation, au(x,t u(x,t) Here u(x,t) > 0 is the concentration of some diffusing substance, the spatial variable is x, time is t and D is a constant called the diffusivity with dimensions [LT-11. We will consider the diffusion equation on a finite spatial domain (0<x< 1) and an infinite time horizon (t > 0). To solve the diffusion equation we must include appropriate boundary conditions and an initial condition: Boundary condition atx 0: u(0,t) 0 for t > 0, ● Boundary condition at x = 1: u (1,1) = 1 for t > 0, and . Initial condition at t 0: u(x,0) 0 for 0 < x < 1 To solve the diffusion equation we will use the Laplace transform of u(x,t), which is defined to be where w(x;s) is a function of one variable, x, and s is the Laplace transform parameter which we think of as an parameter chosen such that the im key point of using the Laplace transform is that w(x;s) is a function of one variable and u(x,t) is a function of two variables. proper integral converges. The 1. By taking the Laplace transform of all appropriate quantities, show that w(x;s) is gov- erned by = sw(x; s), for 0 < x < 1 2. Solve the second order ordinary differential equation for w(x;s), applying appropri- ately transformed boundary conditions, to show that sinh(xVs/D) w(x; s) = s sinh(VST )

Answer 1

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