I’m currently studying nonlinear second-order ordinary differential equations that arise from reaction-diffusion models such as the Fisher and Nagumo equations. I’ve been applying Lie symmetry methods and reduction techniques to find exact solutions, but I’m now exploring additional analytical approaches — no numerical methods.
If you know of powerful methods (e.g. integrating factor techniques, phase plane analysis, transformations, or ansatz-based approaches) or can share relevant papers, I would really appreciate your help. Thank you!
Masindi Phillipos you can use ODE45 function(MATLAB) to resolve your equations.
There are several options available. The most common option is the solution by the power series method. However, many times, especially in cases like the Fisher equation, this gives only a few eigenvalues. Therefore, sometimes, converting the equation to some standard form of ODE (such as Bessel function, Kummer function, etc.) gives better results. I don't know your boundary conditions, so I suggest you read the appendixes of my thesis "Mathematical modeling and computer simulation of membrane processes involving synthetic and natural membranes" (1995), which is available on ResearchGate.
My ODEs don't have boundary conditions or Initial conditions.
I tried Solving them using ADM of which my solutions are in series form, so I don't if I'm doing the right thing by finding my solutions in series form.
Oh,
I don't know the exact form of your equation, however, in general, you can transform your equation into any other well-known function by substitution, but this is an art, only a veteran mathematician can guide you (not me).
Or, you develop the power series solution (assuming, the dependent variable y follows the relation y=a0+a1.x+a2.x^2+... having infinite number of terms). Since your equation is of 2nd order, the solution will contain two arbitrary constants (may be in terms of the first two coefficients, a0 and a1 of the series). A careful analysis may lead to transform your series into some other known functions, or, it can be used as it is (this will become your own function).
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