1. Consider the following advection equation: u+au = 0 _u(x,0) = {-4(x+8) (x + 9), if-9x-8...

 

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1. Consider the following advection equation: u+au = 0 _u(x,0) = {-4(x+8) (x + 9), if-9x-8 otherwise. Use any of the finite difference methods: Upwind, Lax-Friedrich, Lax-Wendroff, Leapfrog to solve the problem for -10 x-10 using a grid where r = = 0.95, a = 1. Plot your solution at time t = 10. 2. Consider the following equation du t subject to the conditions = Pu x z (0,1), t> 0, u(x, 0) u(0. t) = 0 = u(1, t), t20 u(x,0) = x(1-x), 0x1. Discretize the given problem both with either explicit or implicit finite difference schemes. Implement these schemes through MATLAB to find the numerical solution of the given problem. Plot your solutions at t = 0.4 and compare them with the exact solution given by u(x, t) = x(1-x)e-t. 3. Consider the Klein-Gordon equation Pu Fu x t along with the boundary and initial conditions given as u(0,t) = u(1, t) = 0, t 0. +4u, 0 0, Cos - {1 (1-COM (-)). XOSES. if a 2 a 0, otherwise. u(x,0) = 0, 0x 1. Determine and plot the numerical solution up to t= 1.7. Take a = 0.09. 1. Consider the following advection equation: u+au = 0 _u(x,0) = {-4(x+8) (x + 9), if-9x-8 otherwise. Use any of the finite difference methods: Upwind, Lax-Friedrich, Lax-Wendroff, Leapfrog to solve the problem for -10 x-10 using a grid where r = = 0.95, a = 1. Plot your solution at time t = 10. 2. Consider the following equation du t subject to the conditions = Pu x z (0,1), t> 0, u(x, 0) u(0. t) = 0 = u(1, t), t20 u(x,0) = x(1-x), 0x1. Discretize the given problem both with either explicit or implicit finite difference schemes. Implement these schemes through MATLAB to find the numerical solution of the given problem. Plot your solutions at t = 0.4 and compare them with the exact solution given by u(x, t) = x(1-x)e-t. 3. Consider the Klein-Gordon equation Pu Fu x t along with the boundary and initial conditions given as u(0,t) = u(1, t) = 0, t 0. +4u, 0 0, Cos - {1 (1-COM (-)). XOSES. if a 2 a 0, otherwise. u(x,0) = 0, 0x 1. Determine and plot the numerical solution up to t= 1.7. Take a = 0.09.

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