9) Solve the following partial differential equation au a2u ax2 n(0, t) = u(2, t)-0 t > 0 (x, 0) = 0 au at It=0 =x(2-x) 9) Solve the following partial differential equation au a2u ax2 n(0, t)...
9. Solve - cos(x) for 0 <x < 27, t > 0 ax2 at2 y(0, t) y(27, t) = 0 for t 0 y(x, 0) y(x.0)= 0 for 0 <x < 27. at Graph the fortieth partial sum for some values of the time. 11. Solve the telegraph equation au A Bu= c2- at ax2 at2 for 0 x < L, t > 0. A and B are positive constants The boundary conditions are u(0, t) u(L, t)=0 for t...
3. Consider a Laplacian equation in a two-dimensional Cartesian coordinate as a2u + ax2 =0 in 0 < x <a, 0 <y <b ay2 Associate with the boundary conditions of 0ys b at x= 0, u = 0 0Sysb u = f(y) at x = a, 0 x a 0 = n at y 0, at y b, 0 x s a u = 0 (20 points) Find the solution of u(x, y) Reading assignment: Sect. 12.6 of the textbook...
4. (50 pts) Consider the following partial differential equation: 1du au Ət22 Ətər2 (BC) u7,t) = 0 20,t) = 0 0 <t (IC) u(3,0) = 0 0 <r <a Follow the steps below to solve it: (a) (8 pts) Separate variables as u(x,t) = X(2)T(t) to derive the following differential equations for X and T, with an unknown parameter 1: T" - T' + XT = 0, X" + 1X = 0.
Consider the following partial differential equation. au, au ax? + = u ay? Identify A, B, and C in the above equation and use them to calculate the following. B2 - 4AC = -1 + u X Classify the given partial differential equation as hyperbolic, parabolic, or elliptic. O hyperbolic parabolic elliptic
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
2. Solve the following partial differential equation using Laplace transform. Express the solution of u in terms of t&x. alu at2 02u c2 2x2 u(x,0) = 0 u(0,t) = f(t) ou = 0 == Ot=0 lim u(x, t) = 0
The function u(x, t) satisfies the partial differential equation with the boundary conditions u(0,t) = 0 , u(1,t) = 0 and the initial condition u(x,0) = f(x) = 2x if 0<x<} 2(1 – x) if}<x< 1 . The initial velocity is zero. Answer the following questions. (1) Obtain two ODES (Ordinary Differential Equations) by the method of separation of variables and separating variable -k? (2) Find u(x, t) as an infinite series satisfying the boundary condition and the initial condition.
Problem 2: Consider the following differential equation: 0 and with u = e-31. Solve for x(t) using with initial conditions x(0)-x(0) Laplace transforms.
5. (15 %) (a) Solve the partial differential equation u,-ku +cu,, u(x,0)-f(x) k.c are constants and ks0 by taking Fourier transform. (b)Let f(x)-ebe the normal probability density function Find the /(a), and the dispersion of Δ/ , Δ 5. (15 %) (a) Solve the partial differential equation u,-ku +cu,, u(x,0)-f(x) k.c are constants and ks0 by taking Fourier transform. (b)Let f(x)-ebe the normal probability density function Find the /(a), and the dispersion of Δ/ , Δ
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) =...