Consider the equation 3. 1 x < 0 x〉0 u(r, 0) = Find the solution u(x, t) and draw the characteri...
Problem 1. Consider the nonhomogeneous heat equation for u(r, t) subject to the nonhomogenoous boundary conditions u(0, t) 1, u(r, t) 0, t>o and the initial condition u(, 0)in() Find the solution u (z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution t) (b) Den ote u(x, t)-u(x, t)-ue(x). Derive the IBVP for the function u(x,t). (c) Find v(x, t) (d) Find u(x,t) Problem 1. Consider the nonhomogeneous heat equation for u(r, t) subject...
Problem 1. Consider the nonhomogeneous heat equation for u(,) subject to the nonhomogeneous boundary conditions 14(0,t) 1, u(r,t)-0,t> and the initial condition the solution u(x, t) by completing each of the following steps (a) Find the equilibrium temperature distribution u ( (b) Denote v, t)t) - u(). Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t) Problem 1. Consider the nonhomogeneous heat equation for u(,) subject to the nonhomogeneous boundary conditions 14(0,t) 1, u(r,t)-0,t>...
(1) Find the solution for each of the following BVP for the heat equation at)u(r, t)0 (r, t) E (0, 20) x (0, co) (0, t) 0 u(20, t)0 u(r, 0) f(r) E [0, 10 E (10, 20 t > 0 1 where f(r) a. t > 0 (1) Find the solution for each of the following BVP for the heat equation at)u(r, t)0 (r, t) E (0, 20) x (0, co) (0, t) 0 u(20, t)0 u(r, 0) f(r)...
(10') Consider the equation Cut #cut = 1, 2 & R. (a) Find the characteristic curves. Sketch two curves on the t-x plane. (b) Define the characteristic coordinates. (c) Solve the equation with initial condition u(x,0) = x2 + 1.
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0 1. Consider the Partial Differential Equation ot u(0,t) =...
3. Consider the non homogeneous heat equation ut- urr+ 1 with non homogeneous boundary conditions u(0. t) 1, u(1t) (a) Find the equilibrium solution ueqx) to the non homogeneous equation. (b) The solution w(r, t) to the homogenized PDE wt-Wra, with w(0,t,t)0 1S -1 Verify that ugen(x, t)Ue(x) +w(x, t) solves the full PDE and BCs (c) Let u(x,0)- f(x) - 2 - ^2 be the initial condition. Find the particular solution by specifying all Fourier coefficients 3. Consider the...
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t) by completing each of the following steps Find the equilibrium temperature distribution we r) Find th (b) Denote v, t)t) - ()Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t) Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary...
(3) Solve the following BVP for the Wave Equation using the Fourier Series solution formulac (3a2 u(r, t) 0 u(0, t)0 u(T, t) 0 u(r, 0) sin(x)2sin(4r) 3sin(8r) (r, 0) 10sin(2x)20sin (3r)- 30sin (5r) (r, t) E (0, ) x (0, 0o) t >0 t > 0 1 (3) Solve the following BVP for the Wave Equation using the Fourier Series solution formulac (3a2 u(r, t) 0 u(0, t)0 u(T, t) 0 u(r, 0) sin(x)2sin(4r) 3sin(8r) (r, 0) 10sin(2x)20sin (3r)-...
1. Consider the insulated heat equation up = cum, 0 <r<L, t > 0 u (0,t) = u (L, t) = 0, t > 0 u(x,0) = f(2). What is the steady-state solution? 2. Solve the two-dimensional wave equation (with c=1/) on the unit square (i.e., [0, 1] x [0,1) with homogeneous Dirichlet boundary conditions and initial conditions: (2, y,0) = sin(x) sin(y) (,y,0) = sin(x). 3. Solve the following PDE: Uzr + Uyy = 0, 0<<1,0 <y < 2...
Consider the partial differential equation, with the initial condition: 1 2yuz + 3x?uy = 9x?y?, u(x,0) = x3 + 1 Find the characteristic curves and the orthogonal trajectories and sketch both on the same graph. Find a solution of the partial differential equation with the given initial con- dition valid in the first quadrant of the (x, y)-plane. Is this solution unique? Explain.