heat and mass transfer problem nonlinear differential equations
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The code is running, the graph is also came. But it does not give the convergence for the curves s,
, θ, and f.
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1604761/image.png)
fluid_Sol_bvp4c2
function fluid_Sol_bvp4c2
Pr = 10;
Le = 10;
Nr = 0.5;
Nb = 0.5;
Nt = 0.5;
% Defining parameters
solinit = bvpinit(linspace(0, 10, 100), [0 0 1 1 0 1 0]);
sol = bvp4c(@bvp2D, @bc2D, solinit);
x = sol.x;
y = sol.y
% Plotting of the functions
figure(1)
plot(x, y(1,:), 'linewidth', 1), grid
hold on
xlabel('\eta', 'fontweight', 'bold', 'fontsize', 16)
ylabel('s(\eta)', 'fontweight', 'bold', 'fontsize', 16)
hold off
% Plotting of the velocity
figure(2)
plot(x, y(2,:), 'linewidth', 1), grid
hold on
xlabel('\eta', 'fontweight', 'bold', 'fontsize', 16)
ylabel('s^{\prime}(\eta)', 'fontweight', 'bold', 'fontsize', 16)
hold off
% Plotting of theta
figure(3)
plot(x, y(4,:), 'linewidth', 1), grid
hold on
xlabel('\eta ', 'fontweight', 'bold', 'fontsize', 16)
ylabel('\theta ', 'fontweight', 'bold', 'fontsize', 16)
hold off
% Plotting of the fun
figure(4)
plot(x, y(6,:), 'linewidth', 1), grid
hold on
xlabel('\eta', 'fontweight', 'bold', 'fontsize', 16)
ylabel('f(\eta)', 'fontweight', 'bold', 'fontsize', 16)
hold off
% Residual of the boundary conditions
function res = bc2D(y0, yinf)
res = [y0(1); % y1 to start at 0
y0(2); % y2 to start at 0
y0(4) - 1; % y4 to start at 1
y0(6) - 1; % y6 to start at 1
yinf(2) - 0; % desire y2 to converge to 0
yinf(4) - 0; % desire y4 to converge to 0
yinf(6) - 0]; % desire y5 to converge to 0
end
% System of First Order ODEs
function dydx = bvp2D(~, y)
yy1 = - 1/(4*Pr)*(3*y(1)*y(3) - 2*y(2)^2) - y(4) + Nr*y(6);
yy2 = - 3/4*y(1)*y(5) - Nb*y(7)*y(5) - Nt*y(5)^2;
yy3 = - 3/4*Le*y(7) - (Nt/Nb)*yy2;
dydx = [y(2); y(3); yy1; y(5); yy2; y(7); yy3];
end
end
I dont know whether i have to use extend code in bvp4c. Please help me do the needful. Thanks in advance.
1 Comment
Torsten
on 31 Jan 2024
Please include equations and boundary conditions in a mathematical notation so that we can check whether your implementation is correct.
Answers (2)
Sam Chak
on 31 Jan 2024
Hi @uma
I wanted to inform you that I have discovered five sets of Equilibrium Points for the system. However, it appears that the system is unstable. To address this instability, it would be advisable to introduce some manipulatable variables into the system. As a temporary solution, I have incorporated a guess() function into your code. This adjustment has resulted in a different solution when using bvp4c().
%% Equilibrium Points
ye1 = zeros(1, 7);
ye2 = [1, 0, 0, 0, 0, 0, 0];
ye3 = [0, 0, 0, 0.2, 0, 0.4, 0];
ye4 = [0, 0, 0, 0.4, 0, 0.8, 0];
ye5 = [0, 0, 0, 0.6, 0, 1.2, 0];
fluid_Sol_bvp4c2
function fluid_Sol_bvp4c2
Pr = 10;
Le = 10;
Nr = 0.5;
Nb = 0.5;
Nt = 0.5;
% Defining parameters
a = 0;
b = 10;
xmesh = linspace(a, b, 101);
solinit = bvpinit(xmesh, @guess);
sol = bvp4c(@bvp2D, @bc2D, solinit);
% solinit = bvpinit(linspace(0, 10, 100), [0 0 1 1 0 1 0]);
% sol = bvp4c(@bvp2D, @bc2D, solinit);
x = sol.x;
y = sol.y;
% Plotting of the functions
figure(1)
plot(x, y(1,:), 'linewidth', 1), grid
hold on
xlabel('\eta', 'fontweight', 'bold', 'fontsize', 16)
ylabel('s(\eta)', 'fontweight', 'bold', 'fontsize', 16)
hold off
% Plotting of the velocity
figure(2)
plot(x, y(2,:), 'linewidth', 1), grid
hold on
xlabel('\eta', 'fontweight', 'bold', 'fontsize', 16)
ylabel('s^{\prime}(\eta)', 'fontweight', 'bold', 'fontsize', 16)
hold off
% Plotting of theta
figure(3)
plot(x, y(4,:), 'linewidth', 1), grid
hold on
xlabel('\eta ', 'fontweight', 'bold', 'fontsize', 16)
ylabel('\theta ', 'fontweight', 'bold', 'fontsize', 16)
hold off
% Plotting of the fun
figure(4)
plot(x, y(6,:), 'linewidth', 1), grid
hold on
xlabel('\eta', 'fontweight', 'bold', 'fontsize', 16)
ylabel('f(\eta)', 'fontweight', 'bold', 'fontsize', 16)
hold off
% Residual of the boundary conditions
function res = bc2D(y0, yinf)
res = [y0(1); % y1 to start at 0
y0(2); % y2 to start at 0
y0(4) - 1; % y4 to start at 1
y0(6) - 1; % y6 to start at 1
yinf(2) - 0; % desire y2 to converge to 0
yinf(4) - 0; % desire y4 to converge to 0
yinf(6) - 0]; % desire y5 to converge to 0
end
% System of First Order ODEs
function dydx = bvp2D(~, y)
yy1 = - 1/(4*Pr)*(3*y(1)*y(3) - 2*y(2)^2) - y(4) + Nr*y(6);
yy2 = - 3/4*y(1)*y(5) - Nb*y(7)*y(5) - Nt*y(5)^2;
yy3 = - 3/4*Le*y(7) - (Nt/Nb)*yy2;
dydx = [y(2); y(3); yy1; y(5); yy2; y(7); yy3];
end
end
% initial guess
function g = guess(x)
g = [sin(x);
cos(x);
-sin(x);
-cos(x);
sin(x);
cos(x);
-sin(x)];
end
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