Problem 3 Give precise conditions on the initial data 1,2 to ensure the 2 stability of the soluti...
Problem 3 Give precise conditions on the initial data 1,2 to ensure the 2 stability of the solutions to heat PDEs: IR for all t>0 and give an appropriate value of C> 0. Show all steps of the stability proof (as in class) Problem 3 Give precise conditions on the initial data 1,2 to ensure the 2 stability of the solutions to heat PDEs: IR for all t>0 and give an appropriate value of C> 0. Show all steps of...
Problem 3. Given the initial conditions, y(0) from t- 0 to 4: and y (0 0, solve the following initial-value problem d2 dt Obtain your solution with (a) Euler's method and (b) the fourth-order RK method. In both cases, use a step size of 0.1. Plot both solutions on the same graph along with the exact solution y- cos(3t). Note: show the hand calculations for t-0.1 and 0.2, for remaining work use the MATLAB files provided in the lectures Problem...
Let u be the solution to the initial boundary value problem for the Heat Equation, tE (0, o0), те (0, 1); дла(t, г) — 3 Әғu(t, a), with boundary conditions u(t, 0) — 0, и(t, 1) — 0, and with initial condition 0, 1 3 EA 4 u(0, a) f(x) 4. 3 The solution u of the problem above, with the conventions given in class, has the form С сп tn (t) и,(2), u(t, x) - T 1 with the...
#20 please and specifically c.) .... but with the initial conditions only being A= (1,-1) and D=(-1,2). For A, I got x(t)=e^(-4t) and y(t) = -e^(-4t). For D, I got x(t)= 3/4*e^(4t)-7/4*e^(-4t) and y(t)=1/4*e^(4t)+7/4*e^(-4t) 295 3.3 Phase Portraits for Linear Systems with Real Eigenvalues 20. The slope field for the system y 3 dx =2x +6y dt dy = 2x - 2y dt is shown to the right. (a) Determine the type of the equilibrium point at the origin. x...
Solve the heat equation ut = for all time (zero Neumann boundary conditions), if the initial temperature is given by (ax)xsin TX. First, formulate the mathematical problem and complete the three steps as described 10uforarod of length 1 with both ends insulated Mathematical Formulation Step 1 Derive an expression for all nontrivial product (separated) solutions including an eigenvalue problem satisfying the boundary conditions Step 2: Solve the eigenvalue problem Step 3: Use the superposition principle and Fourier series to find...
(1 point) a. Consider the differential equation: d2y 0.16y-0 dt2 with initial conditions dt (0)-3 y(0)--1 and Find the solution to this initial value problem b. Assume the same second order differential equation as Part a. However, consider it is subject to the following boundary conditions: y(0)-2 and y(3)-7 Find the solution to this boundary value problem. If there is no solution, then write NO SOLUTION. If there are infinitely many solutions, then use C as your arbitrary constant (e.g....
3. In class we discussed the heat conduction problem with the boundary conditions a(0, t) 0, t4(1,t)-0, t > 0 and the initial condition u(r,0) f(a) We found the solution to be of the form where (2n-1)n 1,2,3,. TL 20 Now consider the heat conduction problem with the boundary conditions u(0, t) 1,u(T, t)0, t>0 and the initial condition ur,0) 0. Find u(r,t). Hint: First you must find the steady state. 3. In class we discussed the heat conduction problem...
Let u be the solution to the initial boundary value problem for the Heat Equation au(t,) -48Fu(t,), te (0,oo), z (0,5); with boundary conditions u(t,0) 0, u(t,5) 0, and with initial condition 5 15 15 The solution u of the problem above, with the conventions given in class, has the form with the normalization conditions vn(0)-1, u Find the functions vnwn and the constants cn n(t) wnr) Let u be the solution to the initial boundary value problem for the...
Let u be the solution to the initial boundary value problem for the Heat Equation 202u(t, ) te (0, o0) (0,3); дли(t, 2) хе _ with boundary conditions ut, 0) 0 u(t, 3) 0 and with initial condition 3 9 u(0, ar) f(x){ 5, | 4' 4 0, Те The solution u of the problem above, with the conventions given in class, has the form ()n "(2)"п (г)"а "," n-1 with the normalization conditions 3 Wn 2n vn (0) 1,...
Let u be the solution to the initial boundary value problem for the Heat Equation an(t,r)-301a(t, z), te(0,00), z E (0,3); with initial condition 3 0 and with boundary conditions 6xu(t,0)-0, u(t, 3) 0 Find the solution u using the expansion with the normalization conditions vn (0)-1, wn(0) 1 a. (3/10) Find the functionsw with index n1 b. (3/10) Find the functions vn with index n1 Un c. (4/10) Find the coefficients cn, with index n 1 Let u be...