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Question: 1 second order linear pdes of the general form ay...

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1 Second order linear PDEs of the general form ay ax2 дхоу have arbitrary coefficients A, B, C, D, E, F and G that are constant or depend only on the independent variables (a) What condition in respect of these coefficients determines classifi- cation of the solution as elliptic, typical of a steady state? (b) What class of solution is typical with the discriminant A 0? Give a simple example of a physical system that is modelled by this type of solution (c) Identify respectively all regions of the x, y field where the equation 82U дуг is elliptic, parabolic or hyperbolic (ii) We seek a solution over the domain 0 sx S1 of the equation where α and β are positive real constants (a) What is the classification of the solution to this equation? Give a simple example of a physical system that is modelled by this type of solution (b) What auxilliary conditions must be specified to obtain the solution? (c) Let α-1 and β-: 0.5. Use the standard notation U,j-U(x, t) Construct an explicit finite difference formulation of the equation for U,J accurate in space to O(h), where h is the grid separation (d) Given the boundary condition at x 1.0 is given by find the expression for Un.j+1, where Xn 1.0

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