Question

I need to create a MATLAB function, bvp_solve.m, to approximate the solution y(x). The function takes the number of grid points n as an input. The outputs are grid vector x and the solution vector y

ODE: y -y(x - 4) sin(x)+(1 )cos(x) for E [0, 1 BC: y(0) 0, /(0) = -1 (1)sin(1)

%% This is the function i have so far:

function [xi, yi] = bvp_solve(n)
% BVP_SOLVE computes the solution y(x) of a two-point boundary value problem
% using finite difference method (FDM).
% The governing equation is
% y''' = -y + (x - 4) sin(x) + (1 - x)cos(x)
% Boundary conditions: y(0) = 0
% y'(0) = -1
% y'(L) = sin(1)
% The input n is number of grid points used in FDM. The outputs are grid
% vector xi and the solution vector yi.
% Call format: [xi, yi] = bvp_solve(n)

%% Setup grid
L = 1;
h = 1/(n-1);
xi = linspace(0, L, n)';
y3 = @(x, y) -y + (x-4)*sin(x) + (1-x)*cos(x);

%% Construct the linear system M*y = rhs
% Matrix M and rhs
M = zeros(n, n);
rhs = zeros(n,1);

for i = 2:(n-1)
M(i,i-1) = ...;
M(i,i+1) = ...;
M(i,i) = ...;
rhs(i) = ...;
end
%% Enforce boundary conditions:
% y(0) = 0
rhs(1) = 0;
% y'(0) = -1
% y'(1) = sin(1)

%% Solve for yi's:
yi = M\rhs;

end % function bvp_solver

Answer 1

Matlab code:

function [x, y] = ode_three(n)

x = 0:1/n:1;
init = [0 -1 sin(1)];%initial conditions

[x, y] = ode45(@RK4SYSTEM, x, init);%function call to solve
  
function dydt = RK4SYSTEM(x,y)
dydt = zeros(3,1);
dydt(1) = y(2);
dydt(2) = y(3);
dydt(3) = -y(1)+(x-4)*sin(x)+(1-x)*cos(x);
end
disp(y)
disp(x)
  
end

function [x, yl ode three (n) 1 2 0:1/n: 1 [0 -1 sin (1)1;%initial conditions X init 4 _ [x, у] ode45 (@RK4SYSTEM, x, init);%function call to solve 6 7 Command Window ode three (10); 0.84147 0 -1 -0.095643 -0.91152 0.92152 -0.1821 0.96208 -0.81701 0.96422 -0.25897 -0.72038 -0.32623 -0.62539 0.9296 -0.38423 -0.53562 0.86046 0.43364 -0.45436 0.75963 -0.47548 -0.38463 0.63046 -0.32908 0.51104 0.4768 0.54185 -0.28995 0.30293 -0.56964 -0.26902 0.11356 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 st

First column of y is y(x)

Second column of y is y'(x)

Third column of y is y''(x)