1. (Review of initial/boundary value problems for ordinary differential equations) Determine u(x), a the solutions, if...
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.
Numerical Methods for Ordinary Differential Equations: Initial Value Problems 1.10.** Write the IVP x'"' (t) – ax"(t) – bx'(t) – cx = f(t), x(0) = $, s'(0) = n, x"(0) = 5 as a first-order system x'(t) = Ax(t) +g(t). What is the character- istic polynomial of A?
I need help with this PDE problem, part A and D 4.2.3. Write down the solutions to the following initial-boundary value problems for the wave equation in the form of a Fourier series: a) utt = uzz , u(t, 0) = u(t, π) = 0, u(0,x) = 1, ut(0,z) = 0; (b) utt = 2uxx, a(t, 0) = u(t, π) = 0, a(0,x) = 0, ut(0,x) = 1; c) utt = 3uzz , ti(t, 0)=u(t, π)=0, u(0,x) = sin3 x,...
NOTE: • Subject: Numerical Methods for Ordinary Differential Equations: Initial Value Problems. 1. Consider the family of linear multistep methods Un+1 = qUN+ (2(1 – a) f (Un+1) + 3a f (UN) – af (Un-1)). Ppt" (a) Determine the order of accuracy as a function of the parameter a. Find the optimal a to give the highest order of accuracy. Let's call the optimal value Qopt. (b) Is the method with Qopt zero-stable? Is the method with Qopt convergent? Explain...
Question 1 - 16 Consider the following intial-boundary value problem. au au 0<x< 1, 10, at2 ax?' u(0,t) = u(11,t) = 0, 7>0, u(x,0) = 1, 34(x,0) = sin10x + 7sin50x. (show all your works). A) Find the two ordinary differential equations (ODES). B) Solve these two ODES. Show all cases 1 <0, 1 = 0, and > 0 C) Write the complete solution of this initial - boundary value problem.
Differential Equations Series Solutions Near a Ordinary Point find the power series in x for the general solution (1+x^2)y"+2xy-2y=0
Let u be the solution to the initial boundary value problem for the Heat Equation, Otu(t, x) = 2 &n(t, x), ț e (0,00), x e (0,5); with initial condition u(0,xf(x)- and with boundary condition:s Find the solution u using the expansion with the normalization conditions (2n - 1) a. (3/10) Find the functions w, with indexn>1. Wnsin(2n-1)pix/10) b. (3/10) Find the functions v, with indexn > 1. Vnexp(-2(2n-1)pi/10)(2)t) 1. C. (4/10) Find the coefficients cn , with index n...
P3.* Consider the ordinary differential equation: u” + 1 = 0. a) Show that this equation together with the boundary conditions u(0) = 2, u(a) = 0 has no solution. b) Show that this equation together with the boundary conditions u(0) = 2, u(a) = –2 has infinitely many solutions.
write MATLAB scripts to solve differential equations. Computing 1: ELE1053 Project 3E:Solving Differential Equations Project Principle Objective: Write MATLAB scripts to solve differential equations. Implementation: MatLab is an ideal environment for solving differential equations. Differential equations are a vital tool used by engineers to model, study and make predictions about the behavior of complex systems. It not only allows you to solve complex equations and systems of equations it also allows you to easily present the solutions in graphical form....
5. Given the initial-boundary value problem as below: ди ди at +u=k 0<x<1, 1>0, Ox?? Ou -(0,1) Ox Ou (1,t)=0, @x t>0, u(x,0) = x(1 - x) 0<x<1. where k is a non-zero positive constant. (i) By separation of variables, let the solution be in the form u(x,t) = X(x)T(t), show that one can obtain two differential equations for X(x) and T(t) as below: X"-cX = 0 and I' + (1 - ck)T = 0) where c is a constant....