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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:**

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.

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.

Comments

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

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CPallini
20-Jun-24 1:54am

5.

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.

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.

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CPallini
20-Jun-24 1:54am

5.

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.

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