Rewrite the stationary Korteweg-de-Vries equation tu" (3)+au (3) + u(x). (z) = 0. as a first-order...
au + 6. The Korteweg-de Vries equation ди au + 6u 0 ( KV) at ax ar3 is an interesting model partial differential equation because two different physical effects are present: there is an expectation that solutions decay due to the third-order dispersive term; how- ever, the nonlinear term causes waves to steepen. Show that elementary traveling wave solutions of (KdV): u(x,t) = f(x - ct) yields an equation (5) corresponding to conservation of energy: 3(59)2 + fu? - bcf2...
Rewrite the second order differential equation, as a system of a first order differential equation. (see picture) det + sin(a) = 0, 6(0) = el 10) = 0, += [0, 1] 2012 0, t=
1 point) Show that Φ(u, u) (Au + 2, u-u, 7u + u) parametrizes the plane 2x -y-z = 4, Then (a) Calculate Tu T,, and n(u, v). þ(D), where D = (u, u) : 0 < u < 9,0 < u < 3. (b) Find the area of S (c) Express f(x, y, z in terms of u and v and evaluate Is f(x, y,z) ds. (a) Tu n(u,v)- T, (b) Area(S)- (c) JIs f(z, y,2) ds- 1 point)...
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) = u(2, t)-0 t > 0 (x, 0) = 0 au at It=0 =x(2-x)
3) Start with the non-linear force equations and rewrite this as three(3) first-order Ordinary Differential Equations (ODE) or (u, d, w
(1 point) Consider the system of higher order differential equations 2 Rewrite the given system of two second order differential equations as a system of four first order linear differential equations of the formy - P(t)y + g(t). Use the following change of variables y (t) y2(t)y'(t) 3 (t) y(t) у(t) z(t) -y2 4 (1 point) Consider the system of higher order differential equations 2 Rewrite the given system of two second order differential equations as a system of four...
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...
solve the PDE +u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z 382u(t,z), tE (0,oo), E (0,3); with initial condition u(0,x)-f(x)- and with boundary conditions Find the solution u using the expansion u(t,x) n (t) wn(x), with the normalization conditions vn (0)1, Wn (2n -1) a. (3/10) Find the functionswn with index n 1. b. (3/10) Find the functions vn, with index n 1 C. (4/10) Find the coefficients cn , with index n 1. Let...
3. (a) Express the following ordinary differential equation and initial conditions as an autonomous system of first order equations: 2"-223z = 2, '(0)= 1 z(0) 0, (b) Consider the following second order explicit Runge-Kutta scheme written in au- tonomous vector form (y' = f(y)): hf (ynk kihf (yn), k2 yn+1 ynk2. IT Use the second order explicit Runge-Kutta scheme with steplength h compute approximations to z(0.1) and z'(0.1) 0.1 to _ 3. (a) Express the following ordinary differential equation and...