%Code
% Tanh–Coth Method for Solving Nonlinear ODEs
clear; clc; syms U(xi) a0 a1 a2 b1 b2 c xi d1 d2 d3

% Example ODE: U” – U + 2*U^3 = 0
ode = diff(U, xi, 2)*U^2 +(diff(U, xi, 1))^2*(2*U+1)+d1*U^4+d2*U^3;

% Step 1: Assume tanh–coth solution (up to order 2 for example)
U_trial = a0 + a1*tanh(d3*xi) + a2*tanh(d3*xi)^2 + b1*coth(d3*xi) + b2*coth(d3*xi)^2;

% Step 2: Substitute the trial function into the ODE
ode_sub = subs(ode, U, U_trial);

% Step 3: Expand and simplify
ode_simplified = simplify(expand(ode_sub));

% Step 4: Collect terms with respect to tanh and coth
% Convert to polynomial form in tanh(xi) and coth(xi)
ode_collected = collect(ode_simplified, [tanh(d3*xi), coth(d3*xi)]);

% Step 5: Equate coefficients of each power of tanh and coth to zero
% (To make the equation identically zero)
coeffs_tanh = coeffs(ode_collected, tanh(xi));
eqns = [];
for k = 1:length(coeffs_tanh)
eqns = [eqns, simplify(coeffs_tanh(k)) == 0];
end

% Step 6: Solve for coefficients
sol = solve(eqns, [a0 a1 a2 b1 b2 d1 d2 d3], ‘IgnoreAnalyticConstraints’, true);

disp(‘Solutions for coefficients:’)
disp(sol)%Code
% Tanh–Coth Method for Solving Nonlinear ODEs
clear; clc; syms U(xi) a0 a1 a2 b1 b2 c xi d1 d2 d3

% Example ODE: U” – U + 2*U^3 = 0
ode = diff(U, xi, 2)*U^2 +(diff(U, xi, 1))^2*(2*U+1)+d1*U^4+d2*U^3;

% Step 1: Assume tanh–coth solution (up to order 2 for example)
U_trial = a0 + a1*tanh(d3*xi) + a2*tanh(d3*xi)^2 + b1*coth(d3*xi) + b2*coth(d3*xi)^2;

% Step 2: Substitute the trial function into the ODE
ode_sub = subs(ode, U, U_trial);

% Step 3: Expand and simplify
ode_simplified = simplify(expand(ode_sub));

% Step 4: Collect terms with respect to tanh and coth
% Convert to polynomial form in tanh(xi) and coth(xi)
ode_collected = collect(ode_simplified, [tanh(d3*xi), coth(d3*xi)]);

% Step 5: Equate coefficients of each power of tanh and coth to zero
% (To make the equation identically zero)
coeffs_tanh = coeffs(ode_collected, tanh(xi));
eqns = [];
for k = 1:length(coeffs_tanh)
eqns = [eqns, simplify(coeffs_tanh(k)) == 0];
end

% Step 6: Solve for coefficients
sol = solve(eqns, [a0 a1 a2 b1 b2 d1 d2 d3], ‘IgnoreAnalyticConstraints’, true);

disp(‘Solutions for coefficients:’)
disp(sol) %Code
% Tanh–Coth Method for Solving Nonlinear ODEs
clear; clc; syms U(xi) a0 a1 a2 b1 b2 c xi d1 d2 d3

% Example ODE: U” – U + 2*U^3 = 0
ode = diff(U, xi, 2)*U^2 +(diff(U, xi, 1))^2*(2*U+1)+d1*U^4+d2*U^3;

% Step 1: Assume tanh–coth solution (up to order 2 for example)
U_trial = a0 + a1*tanh(d3*xi) + a2*tanh(d3*xi)^2 + b1*coth(d3*xi) + b2*coth(d3*xi)^2;

% Step 2: Substitute the trial function into the ODE
ode_sub = subs(ode, U, U_trial);

% Step 3: Expand and simplify
ode_simplified = simplify(expand(ode_sub));

% Step 4: Collect terms with respect to tanh and coth
% Convert to polynomial form in tanh(xi) and coth(xi)
ode_collected = collect(ode_simplified, [tanh(d3*xi), coth(d3*xi)]);

% Step 5: Equate coefficients of each power of tanh and coth to zero
% (To make the equation identically zero)
coeffs_tanh = coeffs(ode_collected, tanh(xi));
eqns = [];
for k = 1:length(coeffs_tanh)
eqns = [eqns, simplify(coeffs_tanh(k)) == 0];
end

% Step 6: Solve for coefficients
sol = solve(eqns, [a0 a1 a2 b1 b2 d1 d2 d3], ‘IgnoreAnalyticConstraints’, true);

disp(‘Solutions for coefficients:’)
disp(sol) sym a function MATLAB Answers — New Questions

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