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I want to solve a coupled PDE system of three fields $m(x,t)$, $r(x,t)$ and $w(x,t)$, preferably using cpp. The equations are like the following

w= g(w,r,m)
\partial_t r = h(g,w,m)
\partial_t m = J(g,w,m)

where h, J, and g include multiplications of m,r, and w and also derivatives of w,r, and m with respect to x.

What I have tried:

I have solved coupled PDE systems before, but this one is different. And the reason it is different is that I do not have a time derivative (derivative with respect to t) of w in these equations. If  I take the time derivative of the first equation, I get the time derivative of w on left-hand side, but I also have the time derivative of every field on the right-hand side. So I still do not know how to solve this. Could anyone please help me with how to solve this kind of equation? If written code exists for this, I would appreciate it if you could send it to me.
Posted
[no name] 28-Jun-24 4:35am
What is the finite difference method system of equations?
Given that partial derivatives appear to be merely an extension of conventional derivatives, why is the degree of difficulty increasing so io games dramatically?

## Solution 1

Maybe one of the results from this search at DDG will help you : DuckDuckGo — Privacy, simplified.[^]

CPallini 20-Jun-24 1:54am
5.

## Solution 2

Perhaps you should look at such books to see the existence / uniqueness of solution(s), and then formulate it numerically in C++ to solve it.

PARTIAL DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS by Stanley J. Farlow[^]

Elements of Partial Differential Equations (Dover Books on Mathematics) by Ian N Sneddon[^]

Or, look at literature on Finite Difference method / Finite Element method to formulate the problem and seek a solution within the domain of interest. Or also an Integral Equation solution.

All of these depend on the form of the functions - h, J and g. Am not sure whether one can give a generic solution to the problem based on the info you've provided.

v2
CPallini 20-Jun-24 1:54am
5.

## Solution 3

Instead of trying to do it analytically you can try a gradient descent method like Powell's method. Whilst the surface must be differentible (no discontinuites), Powell's method does not require derivatives to be taken.