KdV ODE third-order

It is necessary to find a solution that satisfies the boundary conditions:

u′′′+u′+λu=f.

initial conditions:

u′′(0)=u′(0)+u(0),

u′′(1)=u(1),

u′(1)=u(1),

How to define u and fin matlab?

where λ>0, ;, , and - given numbers.

≤

where C is a positive constant independent of 𝑢 and f

clear;

syms y(x) y0 lambda u

Dy = diff(y);

D2y = diff(y,2);

D2y_2 = diff(y,2);

D3y = diff(y,3);

ode = D3y + lambda*u == y(x);

ySol(x) = dsolve(ode);

Answer 1

[Torsten]

If you don't get a solution with the symbolic approach, you will have to specify the parameters and the function f and try "bvp4c" to solve.

syms u(x) 
%syms f(x)
syms a0 a1 b0 c0 real
%syms lambda
lambda = 1;
f = cos(x);
Du = diff(u);
D2u = diff(u,2);
D3u = diff(u,3);
eqn = D3u+Du+lambda*u==f;
conds = [D2u(0)==a1*Du(0)+a0*u(0),D2u(1)==b0*u(1),Du(1)==c0*u(1)];
solu = dsolve(eqn,conds)

[Athanasios Paraskevopoulos]

As @Torsten said above ,a way to solve this boundary value problem in MATLAB is to use numerical methods such as the bvp4c function. This function is designed for solving boundary value problems for ordinary differential equations numerically. Here is a sample of code

% Define the differential equation system
function dydx = odefun(x, y, lambda)
    dydx = [y(2);
            y(3);
            -y(2) - lambda*y(1) + cos(x)];
end

% Define the boundary conditions
function res = bcfun(ya, yb, a0, a1, b0, c0)
    res = [ya(3) - a1*ya(2) - a0*ya(1); % D2u(0) = a1*Du(0) + a0*u(0)
           yb(3) - b0*yb(1);           % D2u(1) = b0*u(1)
           yb(2) - c0*yb(1)];          % Du(1) = c0*u(1)
end

% Initial guess for the solution
function yinit = guess(x)
    yinit = [0; 0; 0]; % Initial guess
end

% Main script
lambda = 1;
a0 = 1; a1 = 1; b0 = 1; c0 = 1; % Example values, adjust as needed

% Solve the BVP
solinit = bvpinit(linspace(0, 1, 10), @guess);
sol = bvp4c(@(x, y) odefun(x, y, lambda), @(ya, yb) bcfun(ya, yb, a0, a1, b0, c0), solinit);

% Extract and plot the solution
x = linspace(0, 1, 100);
y = deval(sol, x);

figure;
plot(x, y(1, :));
xlabel('x');
ylabel('u(x)');
title('Solution of the differential equation');
grid on;