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Math · Advanced Math
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4 The second order PDE where A, B, C and D are arbitrary constants, can be classified as being either elliptic, parabolic or hyperbolic. (i) Define the three classes in terms of the signature of B2 - 4AC and hence classify each of the following three PDEs: (ii) The one-dimensional diffusion equation is given by where α is the diffusion coefficient. Use the standard finite difference approximations, given on the formulae sheet, together with the notation k At/Ax2, to derive the implicit scheme which approximates this equation (ii) Given that the diffusion equation is to be solved over the range 0 x31 for the temperature distribution along a given steel billet with boundary conditions U(0,t) 0°C and U(1,t) 100°C and initial conditions U(x,0)100x, and assuming that the units have been normalized so that α -1, then, with the aid of a diagram . use the implicit scheme with ΔΧ 0.25 and Δt-0.1 to write down the system of algebraic equations for the temperatures at x0.25, 0.5, 0.75 at time t 0.1; express your equations in the form Axb, and write a scilab program to solve these equations. Your program should define A and b explicitly and then print out the solution x. Note: this is a very short program!

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