How to remove error form code.

Respected Sir/Maam.

I was trying to solve a Nonlinear System ODE by Differential transform method Pade approximation (DTM-PA). Hear It is error " Error using sym/taylor (line 104)" in File "shah (attached)" and "Undefined function 'padeApproximation' for input arguments of type 'symfun'" in file "Asad (attached)" and some errors, which i am not able to rectify. Can you please help. If any information is required, please feel free to ask.

Thanking you in Advance

you help will be highly apprecialbe.

Answer 1

[Walter Roberson]

taylor() accepts at most 3 positional parameters. In order to indicate the order of approximation you must use a name/value pair.

[Syed]

Humble Request sir

kindly edit the code, which you have marked the errors.

[Walter Roberson]

% Define the parameters
M_values = [0, 0.5, 1, 1.5, 2]; % Values of M for which we want to solve the ODE

% Loop over the different values of M
for i = 1:length(M_values)
    M = M_values(i); % Current value of M
    
    % Define the symbolic variables
    syms f(eta) g(eta)
    
    % Define the nonlinear ODEs
    ode1 = diff(f, eta, 3) + f * diff(f, eta, 2) - (diff(f, eta))^2 - M * diff(f, eta) == 0;
    ode2 = diff(g, eta) - 2 * diff(f, eta, 2) - 2 * f * diff(f, eta) == 0;
    
    % Define the initial conditions
    f0 = 0; % f(0) = 0
    g0 = 0; % g(0) = 0
    fp0 = 1; % f'(0) = 1
    
    % Define the Padé approximations
    p = 1; % Padé numerator degree
    q = 1; % Padé denominator degree
    
    % Obtain the Padé approximations for f and g
    avars = sym('a', 1:p);
    bvars = sym('b', 1:p);

    f_pade = taylor(f, eta, 0, 'order', p) * (1 + sum(flip(avars) * (eta^q), 2));
    g_pade = taylor(g, eta, 0, 'order', p) * (1 + sum(flip(bvars) * (eta^q), 2));
    
    % Convert the Padé approximations into functions of eta
    f_func = matlabFunction(f_pade, 'Vars', [eta, avars]);
    g_func = matlabFunction(g_pade, 'Vars', [eta, bvars]);
    
    % Generate data for plotting
    eta_vals = linspace(0, 10, 1000); % Values of eta for which we want to evaluate f and g
    f_vals = f_func(eta_vals);
    g_vals = g_func(eta_vals);
    
    % Plot f and g for the current value of M
    figure;
    plot(eta_vals, f_vals, 'b', 'LineWidth', 1.5);
    hold on;
    plot(eta_vals, g_vals, 'r', 'LineWidth', 1.5);
    legend('f(eta)', 'g(eta)');
    title(['M = ', num2str(M)]);
    xlabel('eta');
    ylabel('f(eta), g(eta)');
    grid on;
end

However, this is not going to work:

• you have not defined any function f or g to be able to evaluate after you have done matlabFunction()
• you have not defined any a1 or b1 values and do not try to pass any additional parameters.

If you were to create symbolic functions instead of using matlabFunction then you could get something that you could evaluate using f_func() and g_func() -- but the results would be in terms of the a* and b* variables, and so would be something you would not be able to plot.

[Syed]

Revered Sir,

Hope you are doing well.

I would like to draw your worthy attention towards error on ODE's with Boundary Conditions (attached).

f and g are define in Boundary conditions

a1 and b1 are define pade approximations p=1 and q=1 or p=2 and q=2

where p denote the numerator degree and q denote denominator degre.

Pade Approximants (PA) are a family of rational functions used to approximate a given function by matching the Taylor series expansion of the function up to a certain order. Pade approximants are widely used in numerical analysis and applied mathematics because they can provide very accurate approximations with a relatively small number of terms in the series.

The general form of a Pade approximant of order (p, q) for a function f(x) is given by:

f(x) ≈ R(x) = (a_0 + a_1 x + a_2 x^2 + ... + a_p x^p) / (1 + b_1 x + b_2 x^2 + ... + b_q x^q)

where the coefficients a_i and b_i are determined by matching the Taylor series expansion of f(x) up to order p+q.

To formulate the PA, we first expand the function f(x) as a Taylor series:

f(x) = a_0 + a_1 x + a_2 x^2 + ... + a_n x^n + O(x^(n+1))

Then we substitute this expansion into the general form of the Pade approximant and match the coefficients of like powers of x. This results in a set of linear equations that can be solved for the coefficients a_i and b_i. The resulting Pade approximant is then used as an approximation for the original function f(x).

I tried my level best but all in vain.

I humbly request your honor to remove the error and solve the said issue.

I would be very helpfull for your help.

Thank you ahead of time.

[Syed]

sir @Walter Roberson, i am waiting for your response

[Walter Roberson]

p = 1; q = 2;

syms a0 b0 x
syms a [1 p]
syms b [1 q]
a = [a0 a];
b = [b0 b];
n = sum(a .* x.^(0:p))

n = 

d = sum(b .* x.^(0:q))

d = 

f(x) = n / d

f(x) = 

You will probably want to look at coeffs for matching coefficients. And you might possibly need numden

[Syed]

yes sir @Walter Roberson and also plot the grahp for given ODE with Boundary conditions by using Differential transform method

[Walter Roberson]

Sorry, I am not willing to learn Differential Transform Method at this time. I am a bit distracted by other things, so ask specific short questions and I might answer, but I am not willing to go through the mathematics of what you are trying to do.

I have shown you how to construct the rational polynomial, and shown you the command needed to match coefficients; you can work from there.

[Syed]

@Walter Roberson

Humble Request sir

Sir i need series solution graph (eq 10 and 17) and attach the file in PDF form (all details are disussed in PDF File)

I humbly request your honor to solve the said issue.

I would be very helpfull for your help.

Thank you ahead of time.