Please provide clear and by step solution Determine the solution to the one-dimensional wave equation where a = 3, L = 10,u(x, 0) = 0 andduƏt-(x,0) = =0,1,0,0≤x≤ 3,3≤ x < 44≤x≤ 10.

Answered: Image /qna-images/answer/dffc889c-df3f-4f93-8508-4bbdc8aa8ef0.jpgclose allclear allclc% Parametersalpha = 3;L = 10;tmax = 10; % Max time for plottingnumPoints = 500;% Define the time and space gridsx = linspace(0, L, numPoints);t = linspace(0, tmax, numPoints);[X, T] = meshgrid(x, t);% Initialize the solution matrixu = zeros(size(X));% Calculate the solutionfor i = 1:length(t) for j = 1:length(x) xt_lower = X(i,j) - alpha * T(i,j); xt_upper = X(i,j) + alpha * T(i,j); if xt_upper < 3 % Case 1: Solution is zero u(i,j) = 0; elseif X(i,j) < 3 && xt_upper >= 3 && xt_upper < 4 % Case 2: Lower part of the piecewise function u(i,j) = 1/(2*alpha) * (xt_upper - 3); elseif X(i,j) < 4 % Case 2: Upper part of the piecewise function u(i,j) = 1/(2*alpha); else % Case 3: Solution is zero u(i,j) = 0; end endend% Plot the solutionfigure;surf(X, T, u);xlabel('Position (x)');ylabel('Time (t)');zlabel('Displacement (u(x, t))');title('Solution of the One-Dimensional Wave Equation');shading interp; % Smooth the color transition on the surfacecolorbar; % Add a color bar