Hi @Muhammad Zeeshan ,

After going through your comments, analysis of your code and documentation provided at the link below.

Since you have to purchase journal and not able to provide full details, this is how I would tackle the fourth-order partial differential equation (PDE) you have described, you need to set up a MATLAB script that incorporates the specified linear and nonlinear operators, boundary conditions, and initial conditions derived from the manuscript you provided. Your problem revolves around fluid dynamics, specifically a flow between parallel walls influenced by oscillatory motion. The behavior of the flow is affected by parameters such as amplitude (Δ) and Reynolds number (R), with chaotic behavior emerging under certain conditions. The PDE is significant as it models these dynamics and serves as a foundation for understanding complex flow phenomena. Below is an updated MATLAB code that defines the necessary parameters and implements a numerical method to solve the given PDE. The code uses finite difference methods or similar approaches for spatial discretization, and can be adjusted based on your specific needs for stability and accuracy.

% Parameters L = 10; % Length of the domain T = 20; % Total time duration Nx = 100; % Number of spatial points Nt = 2000; % Number of time steps dx = L/(Nx-1); % Spatial step size dt = T/Nt; % Time step size

% Physical parameters delta = 0.1; % Amplitude of oscillation Re = 100; % Reynolds number

% Define spatial grid x = linspace(0, L, Nx);

% Initial condition: H(t) H = @(t) 1 + delta * cos(2*t); dHdt = @(t) -2 * delta * sin(2*t); % Derivative of H with respect to t

% Initialize V field V = zeros(Nx, Nt); V(:, 1) = H(0); % Set initial condition at t=0

% Time-stepping loop for n = 1:Nt-1 t = n * dt;

% Compute V at next time step using explicit method (pseudo-code) for i = 2:Nx-1 % Linear operator LV (eq. 2.5) LV = (V(i+1,n) - 2*V(i,n) + V(i-1,n)) / dx^2;

% Nonlinear operator NV (eq. 2.6) NV = (1/H(t)) * (V(i,n) * V(i,n) * V(i,n) - V(i,n)^2);

% Update V based on the PDE V(i,n+1) = V(i,n) + dt * (LV + NV); end

% Apply boundary conditions V(1,n+1) = H(t); % V(0,t) = H(t) V(end,n+1) = 0; % V(L,t) = 0 end

% Visualization of results figure; surf(linspace(0,T,Nt), x, V); xlabel('Time'); ylabel('Position'); zlabel('Velocity Field'); title('Fluid Flow Between Parallel Walls'); colorbar;

Please see attached.

In the above script, parameters are initialized such as the length of the domain, total time duration, number of spatial points, and time. The initial condition for H(t) is set as specified in your context. A finite difference approach is utilized to approximate derivatives in space and time: The linear operator LV is calculated using central differences. The nonlinear operator NV is computed directly from the defined equation. Boundary conditions are applied after updating V at each time step. Finally, the results are visualized in a surface plot showing how velocity changes over time and space.

Make sure that your chosen time step dt and spatial resolution dx satisfy stability criteria for numerical methods employed. Depending on the specifics of your problem and computational resources, you may consider more sophisticated methods such as implicit schemes or spectral methods for better stability in solving higher-order PDEs. Conduct sensitivity analyses on delta and Re to explore different regimes of flow behavior, including stability and chaos.

This code at the moment provides a foundational framework to address the fluid dynamics problem you are investigating while adhering to the details outlined in the manuscript. Adjustments may be needed based on specific requirements or further insights from your research context.