Numerical Analysis
3rd Edition
ISBN: 9780134696454
Author: Sauer, Tim
Publisher: Pearson,
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Chapter 8.2, Problem 3CP
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To find: the solution by taking IVP values by FEM method using in matlab program.
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Find the solution to the wave equation on the half - line:
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Use the d'Alambert formula for the wave equation to find the solution for the given initial conditions.
Consider the wave equation on an infinite line, with u(x, 0) = f (x) defined by f (x) = { 0, for x < 0, x^2, for 0 ≤x ≤1, (2 −x), for 1 ≤x ≤2, 0, for x > 2.
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Chapter 8 Solutions
Numerical Analysis
Ch. 8.1 - Prove that the functions (a) u(x,t)=e2t+x+e2tx,...Ch. 8.1 - Prove that the functions (a) u(x,t)=etsinx, (b)...Ch. 8.1 - Prove that if f(x) is a degree 3 polynomial, then...Ch. 8.1 - Prob. 4ECh. 8.1 - Verify the eigenvector equation (8.13).Ch. 8.1 - Show that the nonzero vectors vj in (8.12 ), for...Ch. 8.1 - Prob. 1CPCh. 8.1 - Consider the equation ut=uxx for 0x1, 0t1 with the...Ch. 8.1 - Prob. 3CPCh. 8.1 - Use the Backward Difference Method to solve the...
Ch. 8.1 - Use the Crank-Nicolson Method to solve the...Ch. 8.1 - Prob. 6CPCh. 8.1 - Prob. 7CPCh. 8.1 - Setting C=D=1 in the population model (8.26), use...Ch. 8.2 - Prove that the functions (a) u(x,t)=sinxcos4t, (b)...Ch. 8.2 - Prove that the functions (a) u(x,t)=sinxsin2t, (b)...Ch. 8.2 - Prove that u1(x,t)=sinxcosct and u2(x,t)=ex+ct are...Ch. 8.2 - Prove that if s(X) is twice differentiable, then...Ch. 8.2 - Prove that the eigenvalues of A in (8.33) lie...Ch. 8.2 - Let be a complex number. (a) Prove that if +1/ is...Ch. 8.2 - Solve the initial-boundary value problems in...Ch. 8.2 - Solve the initial-boundary value problems in...Ch. 8.2 - Prob. 3CPCh. 8.2 - Prob. 4CPCh. 8.3 - Show that u(x,y)=ln(x2+y2) is a solution to the...Ch. 8.3 - Prob. 2ECh. 8.3 - Prove that the functions (a) u(x,y)=eysinx, (b)...Ch. 8.3 - Prove that the functions (a) u(x,y)=exy, (b)...Ch. 8.3 - Prove that the functions (a) u(x,y)=sin2xy, (b)...Ch. 8.3 - Prove that the functions (a) u(x,y)=ex+2y, (b)...Ch. 8.3 - Prob. 7ECh. 8.3 - Show that the barycenter of a triangle with...Ch. 8.3 - Prove Lemma 8.9 .Ch. 8.3 - Prove Lemma 8.10.Ch. 8.3 - Prob. 11ECh. 8.3 - Prob. 12ECh. 8.3 - Prob. 13ECh. 8.3 - Solve the Laplace equation problems in Exercise 3...Ch. 8.3 - Prob. 2CPCh. 8.3 - Prob. 3CPCh. 8.3 - Prob. 4CPCh. 8.3 - Prob. 5CPCh. 8.3 - The steady-state temperature u on a heated copper...Ch. 8.3 - Prob. 7CPCh. 8.3 - Prob. 8CPCh. 8.3 - Solve the Laplace equation problems in Exercise 3...Ch. 8.3 - Solve the Poisson equation problems in Exercise 4...Ch. 8.3 - Solve the elliptic partial differential equations...Ch. 8.3 - Prob. 12CPCh. 8.3 - Prob. 13CPCh. 8.3 - Solve the elliptic partial differential equations...Ch. 8.3 - Prob. 15CPCh. 8.3 - Prob. 16CPCh. 8.3 - For the elliptic equations in Exercise 7, make a...Ch. 8.3 - Solve the Laplace equation with Dirichlet boundary...Ch. 8.4 - Show that for any constant c, the function...Ch. 8.4 - Show that over an interval [ x1,xr ] not...Ch. 8.4 - Prob. 3ECh. 8.4 - Prob. 4ECh. 8.4 - Prob. 5ECh. 8.4 - Prob. 6ECh. 8.4 - Prob. 1CPCh. 8.4 - Prob. 2CPCh. 8.4 - Solve Fishers equation (8.69) with...Ch. 8.4 - Prob. 4CPCh. 8.4 - Solve the Brusselator equations for...Ch. 8.4 - Prob. 6CP
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- Show that the function z = cos(4x + 4ct) satisfies the wave equation ∂2 z/ ∂t2 = c2 (∂2 z /∂x2 ).arrow_forwardShow that f(x, y) =(Aekx + Be-kx )(Ce2ky + De−2ky) is a solution of the wave equation∂2f/∂y2− 4∂2f/∂x2 =0.arrow_forwardsolve the one dimensional wave equation with the boundary conditions and inital conditions as given below: δ2u/δt2 = 1/pi2.δ2u/δx2 u(0,t)= 0, t>0. u(1,t)=0, t>0 u(x,0)= sinππxcosπx, 0<x<1 δu/δt(x,0)=0 0<x<1 using the method of seperation of variablearrow_forward
- for wave equation, seperation of vairables u(x,t)=X=(x)T(t)arrow_forwardFor the wave equation in R3(i.e., (x1 , x2 , x3 ) ∈ R3), let u = u(r, t) be theradial symmetric solution of the systemutt = ∆u, 0 ≤ r < ∞, t > 0u(r, 0) = 1, r ≥ 0ut(r, 0) =2, 0 ≤ r ≤ 1,0, r > 1 Find u(2, 1) and u(2, 2).arrow_forwardConsider the family of functions uc(x,t) of two variables x,t, indexed by the parameter c,uc(x,y)=ln(x+ct)(cos(ct)cos(x)−sen(ct)sin(x)).Determine the value of the parameter c>0 so that the function uc(x,y) is a solution of the wave equationarrow_forward
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