Equation involving Incomplete Gamma function

How to solve this equation using matlab

where and and here m, , are constant.

[Star Strider]

The purpose appears to be solving for m and θ, so code it using ‘b(1)’ for m and ‘b(2)’ for θ, express it as:

fcn = @(b) <expression> - tan(2*b(2));

and use fsolve to solve for the ‘b’ parameters. Choose appropriate non-zero initial parameter estimates.

[Sudhir Sahoo]

Hello thanks for your comment. @Star Strider However, here m is constant and only b and a are function of θ. hence θ is the only variable.

[Star Strider]

O.K. I missed seeing that m was constant. Code the expression and solve it in terms of only θ.

Answer 1

[Nipun]

Hi Sudhir,

I understand that you want to solve the given equation in MATLAB comprising of trigonometric and incomplete gamma functions.

I recommend using the "symbolic Math Toolbox" in MATLAB to create the equation and using a MATLAB solver to solve for "theta". MATLAB supports incomplete gamma functions using an in-built function, called "gammainc".

Here are the steps you may take to slve the given equation:

1. Create symbolic variables
2. Code the equation
3. Use solver to solve for "theta"

I am attaching a code snippet with this answer for your reference:

syms theta m tmin tmax

a = 2*sin(theta).^2
b = (cos(theta)-sin(theta)).^2/2

% Equation (split into two parts)
part1 = (4/(cot(theta)-1).^2)^((m+2)/(m+3))

arg1 = (m+1)/(2*m+6);
arg2 = b*tmin/2;
arg3 = a*tmin/2;
arg4 = b*tmax/2;
arg5 = a*tmax/2;

part2_numerator = gammainc(arg1, arg2) - gammainc(arg1, arg4);
part2_denominator = gammainc(arg1, arg3) - gammainc(arg1, arg5);

part2 = part2_numerator/part2_denominator;

%  creating equation
eqn = part1*part2

%  Solving
ans = solve(eqn, theta);

For more information on the in-built functions, refer to the following MathWorks documentation:

1. Symbolic Toolbox: https://www.mathworks.com/help/symbolic/syms.html#d126e328782
2. Incomplete Gamma Function: https://www.mathworks.com/help/matlab/ref/gammainc.html
3. Equation Solver: https://www.mathworks.com/help/symbolic/sym.solve.html

Hope this helps.

Regards,

Nipun