Facing problems in nonlinear system

%%%% Problem_01 %%%%
%u_t=u_xx+6u(1-u)%
%u(0,t)=(1+e^(-5t))^(-2)%
%u(1,t)=(1+e^(1-5t))^(-2)%
%u(x,0)=(1+e^x)^(-2)%

clc;
clear all;
format short

L = 1; % Length of the rod
T = 0.05; % Total time
Nx = 7; % Number of spatial steps
Nt = 10; % Number of time steps
alpha = 1; % Thermal diffusivity
dx = L / Nx; % Spatial step size
dt = T / Nt; % Time step size
r = alpha * dt / dx^2
u = sym(‘u’, [Nx+1,Nt+1]);

% Define the spatial grid
x = linspace(0, L, Nx+1);
x

% Set initial condition u(x,0)
u(:, 1) = vpa(0.0001679.*(x.^2-x)+0.00002215.*(1.5.*x.^3-1.5.*x),9);

% Set Dirichlet boundary conditions
u(1, 🙂 = boundary_condition_x0(linspace(0, T, Nt+1)); % u(0,t)
u(end, 🙂 = boundary_condition_xL(linspace(0, T, Nt+1)); % u(1,t)
u;

% Initialize the source term matrix
f = [];
R = [];
for n = 1:Nt
for j = 1:Nx+1
f_expr = (-10*(exp(x(j)-5*(n-0.5)*dt))/(1+exp(x(j)-5*(n-0.5)*dt))^3) -(2*exp(x(j)-5*(n-0.5)*dt)/(1+exp(x(j)-5*(n-0.5)*dt)^3)*((1-2*exp(x(j)-5*(n-0.5)*dt)))/(1+ …
exp(x(j)-5*(n-0.5)*dt)))-0.0003358-0.00019935*x(j) +6*((1/(1+exp(x(j)-5*(n-0.5)*dt))^2)-0.0001679*(x(j).^2-x(j))-0.00002215*((3/2)*(x(j).^3-x(j))))*(1-(1/(1+ …
exp(x(j)-5*(n-0.5)*dt))^2) +0.0001679*(x(j).^2-x(j))+0.00002215*((3/2)*(x(j).^3-x(j)))) + 6*0.5*(u(j,n)+u(j,n+1))*(1-0.5*(u(j,n)+u(j,n+1)));
f{j,n} = f_expr;
end
for j = 2:Nx
eq = (1-6*r)*u(j-1, n+1) + (10 + 12*r)*u(j, n+1) + (1 – 6*r)*u(j+1, n+1) == (1 + 6*r)*u(j-1, n)…
+ (10-12*r)*u(j, n) + (1 +6*r)*u(j+1, n) + dt*(f{j-1,n} +10*f{j,n} + f{j+1,n});
eqs(n,j-1) = eq;
end
% disp("Equations before solving:");
% disp(vpa(eqs(n, :), 6));
vsol = vpasolve(eqs(n,:));
R = struct2cell(vsol);
for j = 2:Nx
u(j,n+1) = min(abs(R{j-1}));
end
end
vpa(u,9);

esol = @(x,t) (1+exp(x-5*t))^(-2);
exact_sol = [];
Compact_sol = [];
Error_Compact = [];
for n = 2:Nt+1
for j = 2:Nx+1
exact_sol(j,n) = esol((j-1)*dx,(n-1)*dt);
Compact_sol(j,n) = u(j,n);
Error_Compact(j,n) = abs(exact_sol(j,n)-Compact_sol(j,n));
end
end
n = 11; % Choose any specific value of n (1 to 10)
j_values = 1:Nx+1;
u_values = u(j_values, n);
exact_val = exact_sol(j_values, n);
Compact_val = Compact_sol(j_values,n);
Compact_error_val = Error_Compact(j_values, n);

Table = table(u_values, exact_val,Compact_val,Compact_error_val, …
‘VariableNames’, {‘E(i,j)’, ‘Exact_Solution’,’Compact_Solution’,’Compact_Error’})

function bc_x0 = boundary_condition_x0(t)% E(0,t)
bc_x0 = zeros(size(t));
end

function bc_xL = boundary_condition_xL(t)% E(1,t)
bc_xL = zeros(size(t));
end%%%% Problem_01 %%%%
%u_t=u_xx+6u(1-u)%
%u(0,t)=(1+e^(-5t))^(-2)%
%u(1,t)=(1+e^(1-5t))^(-2)%
%u(x,0)=(1+e^x)^(-2)%

clc;
clear all;
format short

% Define the spatial grid
x = linspace(0, L, Nx+1);
x

% Set initial condition u(x,0)
u(:, 1) = vpa(0.0001679.*(x.^2-x)+0.00002215.*(1.5.*x.^3-1.5.*x),9);

% Set Dirichlet boundary conditions
u(1, 🙂 = boundary_condition_x0(linspace(0, T, Nt+1)); % u(0,t)
u(end, 🙂 = boundary_condition_xL(linspace(0, T, Nt+1)); % u(1,t)
u;

Table = table(u_values, exact_val,Compact_val,Compact_error_val, …
‘VariableNames’, {‘E(i,j)’, ‘Exact_Solution’,’Compact_Solution’,’Compact_Error’})

function bc_x0 = boundary_condition_x0(t)% E(0,t)
bc_x0 = zeros(size(t));
end

function bc_xL = boundary_condition_xL(t)% E(1,t)
bc_xL = zeros(size(t));
end %%%% Problem_01 %%%%
%u_t=u_xx+6u(1-u)%
%u(0,t)=(1+e^(-5t))^(-2)%
%u(1,t)=(1+e^(1-5t))^(-2)%
%u(x,0)=(1+e^x)^(-2)%

clc;
clear all;
format short

% Define the spatial grid
x = linspace(0, L, Nx+1);
x

% Set initial condition u(x,0)
u(:, 1) = vpa(0.0001679.*(x.^2-x)+0.00002215.*(1.5.*x.^3-1.5.*x),9);

% Set Dirichlet boundary conditions
u(1, 🙂 = boundary_condition_x0(linspace(0, T, Nt+1)); % u(0,t)
u(end, 🙂 = boundary_condition_xL(linspace(0, T, Nt+1)); % u(1,t)
u;

Table = table(u_values, exact_val,Compact_val,Compact_error_val, …
‘VariableNames’, {‘E(i,j)’, ‘Exact_Solution’,’Compact_Solution’,’Compact_Error’})

function bc_x0 = boundary_condition_x0(t)% E(0,t)
bc_x0 = zeros(size(t));
end

function bc_xL = boundary_condition_xL(t)% E(1,t)
bc_xL = zeros(size(t));
end nonlinear, equation system, solve MATLAB Answers — New Questions

Cart

Cart