clc;clear all;close all;
A=diag(-3.5*ones(1,4))+diag(1.25*ones(1,3),-1)+diag(1.25*ones(1,3),1);
B=-[sin(0.2*pi);sin(0.4*pi);sin(0.6*pi);sin(0.8*pi)];
X1=inv(A)*B
X2=inv(A)*(-1*X1)
Matlab result
X1 =
0.3978
0.6437
0.6437
0.3978
X2 =
0.2693
0.4357
0.4357
0.2693
>>
au ca, t) - azu (nt) = 0 at an² ocael ost uncoot)= ullit)=0 linio): Sincan)...
USING ( Finite Difference Method for PDEs ) au ca, t) - azu (nt) = 0 at an² ocael ost uncoot)= ullit)=0 linio): Sincan) 418,0:5)=? lei hao.2, k= 0.25
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)
b) Consider the wave equation azu azu at2 0 < x < 2, t>0, ar2 with boundary conditions u(0,t) = 0, u(2, t) = 0, t> 0, and initial conditions u(x,0) = x(2 – x), ut(x,0) = = 0, 0 < x < 2. Use the method of separation of variables to determine the general solution of this equation. (15 marks)
P3. In the circuit shown, let DUO 0, -00<t<0 v(t) = { 1, Ost<10s at (10, 1055t<00 (a) Find the energy stored in the capacitor as a function of t, for 0 st 50. (b) Find the energy delivered by the source as a function of t, for 0 stsoo. va) 0.1F 322 Figure P4.7
a) Use the d'Alembert solution to solve au au - <r< ,t> 0, at2 48,2 ux,0) = cos 3x, u(,0) = 21 b) Consider the heat equation диди 0<x<1, t > 0, at ax? with boundary conditions uz (0,t) = 0, uz(1,t) = 0, > 0, and initial conditions u(x,0) = { 0, 2.0, 0<r < 0.5, 0.5 <<1. Use the method of separation of variables to solve the equation.
Find the solution to the heat equation on the infinite domain ∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,0,|x|<1|x|>1.∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,|x|<10,|x|>1. in terms of the error function. Q1 (10 points) Find the solution to the heat equation on the infinite domain azu ди at k -00<x<0, t>0, ar2 u(x,0) (X, 1x < 1 10, [] > 1. in terms of the error function. + Drag and drop your files or click to browse...
Solve the heat flow problem: ot (x, t) au au (x, t) = 2 (x, t), 0 < x <1, t > 0, a x2 uz(0,t) = uz(1, t) = 0, t> 0, u(a,0) = 1 + 3 cos(TTX) – 2 cos(31x), 0<x< 1.
P3. Solve the equation au(t, x) = kazu(t, x)-γυ(t, x) a(0, r) = f(x) f or-00 < x < oo with f E L(R), where k > 0 and γ E R. P3. Solve the equation au(t, x) = kazu(t, x)-γυ(t, x) a(0, r) = f(x) f or-00
3. Find the Laplace transform off, where f(t) = 3 + 2 if Ost <3, f(t) = 0 if 3 st < 6 and f is periodic with period 6. 4. Solve y" - 16y = 40e4t y(0) = 5, y(0) = 9 using the Laplace transform.
au (x, at2 a (2,t), 0 < x < 57, to ac2 u(0,t) = 0, u(57, t) = 0, t>0, u(3,0) = sin(4x), ut(x,0) 4 sin(5x), 0 < x < 57. u(x, t) =