how to solve differential equation

I want to solve differential equation plz help

du/dt+u*du/dx+v*du/dy+w*du/dz=-1/p*dp/dx+r(d^2u/dx^2+d^2u/dy^2+d^2u/dz^2)

the values of v,w,du/dt,dv/dt.dw,dt will be determined nu user

plz tell how to implement

Answer 1

[Gautam]

Hello, FIR

Here's a basic workflow that you can follow:

1. Discretize the equation.
2. Set up initial and boundary conditions.
3. Let user define ( v, w, dt ) (and others as needed).
4. Iterate over time using numerical integration (Euler, Runge-Kutta, etc.).
5. Visualize or output the results.

You can try out something like this

 % Parameters
Nx = 100;              % Number of spatial points
L = 1;                 % Length of domain
dx = L/(Nx-1);         % Spatial step
dt = 0.001;            % Time step (user-defined)
Nt = 200;              % Number of time steps
u = zeros(Nx,1);       % Velocity u
p = zeros(Nx,1);       % Pressure
rho = 1;               % Density
r = 0.01;              % Viscosity

% Initial conditions
u(:) = 0;              % or user-defined
p(:) = 0;              % or user-defined

% Main time-stepping loop
for n = 1:Nt
    u_old = u;
    p_old = p;
    for i = 2:Nx-1
        du_dx = (u_old(i+1) - u_old(i-1))/(2*dx);
        d2u_dx2 = (u_old(i+1) - 2*u_old(i) + u_old(i-1))/(dx^2);
        dp_dx = (p_old(i+1) - p_old(i-1))/(2*dx);

        % specify v, w, du/dt, dv/dt, dw/dt as needed
        v = 0; % or user-defined
        w = 0; % or user-defined

        % Time derivative (explicit Euler)
        u(i) = u_old(i) + dt * ( ...
            - u_old(i) * du_dx ...
            - (1/rho) * dp_dx ...
            + r * d2u_dx2 ...
        );
    end
    % Boundary conditions
    u(1) = 0; u(end) = 0; % Example BCs
end

plot(linspace(0,L,Nx), u)
xlabel('x'); ylabel('u');
title('Velocity profile');