Hello how are you doing ? Please the questions is on the image and all the details thanks you so much for your help 1. Problem: Numerical Simulation of One-Dimensional Convection-DiffusionProblem using Finite Volume MethodControl equation:with F=40x and [=1.d (Fo) d do(Tdx dx dxBoundary conditions: at x=0, D=1; atx=1, p=0.Compute the distribution of O between x=0 and x=1 using the Finite VolumeMethod.2. Requirements:(1) Derive the discretizing and solving formulas using the Finite Volume Method,and prepare a report in the required format.(2) Implement the source code in Fortran or C language(3) Compare the computed result with the exact solution, and include acomparison graph in your report.e 20x²The exact solution is: eSeSe dx-20x-20x²[1-dx-1

Answered: Image /qna-images/answer/13b8b848-e1b1-4e9b-982c-93e4c54e2eec.jpgclose all clear allclc% ParametersGamma = 1; % Diffusion coefficient, Gamma = 1L = 1; % Length of the domainN = 100; % Number of control volumesdx = L / N; % Width of each control volumex = linspace(dx/2, L - dx/2, N)'; % Cell-centered grid points% Define the convection coefficient function FF = @(x) 40 * x; % Convection coefficient is a function of x% Preallocate arrays for the coefficients and solutionphi = zeros(N, 1); % Solution arrayaE = zeros(N, 1); % Coefficients for the east (right) side of control volumesaW = zeros(N, 1); % Coefficients for the west (left) side of control volumesaP = zeros(N, 1); % Coefficients for the control volume itself% Boundary conditionsphi(1) = 1; % Left boundary condition, phi(0) = 1phi(N) = 0; % Right boundary condition, phi(L) = 0% Iterative solver setupmaxIter = 10000; % Maximum number of iterationstolerance = 1e-6; % Convergence toleranceiter = 0; % Iteration counterresidual = inf; % Initialize residual% Iterative solver using while loopwhile residual > tolerance && iter < maxIter phi_old = phi; % Store the old values of phi for convergence check % Loop over all cells to update solution for i = 2:N-1 % Convection term using upwind scheme Fw = max(F(x(i-1)),0); % Upwind face value for F at west face Fe = max(-F(x(i)),0); % Upwind face value for F at east face % Compute coefficients aW(i) = Gamma / dx + Fw; aE(i) = Gamma / dx + Fe; aP(i) = aW(i) + aE(i) + (F(x(i)) - F(x(i-1))); % Total flux out of the control volume % Update solution phi for internal nodes phi(i) = (aW(i)*phi(i-1) + aE(i)*phi(i+1)) / aP(i); end % Enforce boundary conditions phi(1) = 1; % Left boundary phi(N) = 0; % Right boundary % Calculate the residual for this iteration residual = norm(phi - phi_old, inf); % Increment iteration counter iter = iter + 1;end% Check if we've convergedif residual < tolerance fprintf('Converged in %d iterations\n', iter);else fprintf('Max iterations reached without convergence. Final residual: %g\n', residual);end% Output the result of the iterationif residual < tolerance fprintf('Converged in %d iterations\n', iter);else fprintf('Max iterations reached without convergence. Final residual: %g\n', residual);end% Plot the numerical solutionplot(x, phi, 'b-', 'LineWidth', 2);hold on; % Hold the plot for overlaying the exact solution% Calculate the exact solution at the cell centers for comparisonphi_exact = zeros(N, 1);for i = 1:N xi = x(i); phi_exact(i) = exp(20 * xi^2) * (1 - integral(@(x) exp(-20 * x.^2), 0, xi) / integral(@(x) exp(-20 * x.^2), 0, 1));end% Plot the exact solutionplot(x, phi_exact, 'r--', 'LineWidth', 2);legend('Numerical Solution', 'Exact Solution', 'Location', 'northeast');xlabel('x', 'FontSize', 12, 'FontWeight', 'bold');ylabel('\phi', 'FontSize', 12, 'FontWeight', 'bold');title('Comparison of Numerical and Exact Solutions', 'FontSize', 14, 'FontWeight', 'bold');% Enhance the plotset(gca, 'FontSize', 12, 'LineWidth', 1.5); % Set the properties of the axisgrid on; % Turn on the gridbox on; % Enclose the plot in a box% Optionally, set the limits of the axes if necessaryxlim([0, L]); % Set the limit for the x-axisylim([0, max([phi; phi_exact])]); % Set the limit for the y-axis based on the max value of phihold off; % Release the plot