![]() But not knowing the equation(s) being solved for: my head is scratching. It's easy to turn a partial equation into a (set of) ordinary differential equation knowing the original equation. If one PDE is a gradient on a surface any (directional derivative allowed) is possible. I'd say PDE can be reduced to several ODE, because you can introduce trace equations so each is no longer partial. seems to show principles and perhaps parts of proofs for systems of PDE: but I think in a limited way (a few forms where they can be solved). It requires Cauchy-Riemann considerations and "sacrifices" of some areas to arrive at the form. Laplace's equation in cylindrical coordinates looks "odd" and it is: it is formed not by simple translation of coord by conditions of preserving IMZ in such ways as Laplace needed (the handling of singularities needed for his solutions). Wikipedia says Cauchy-Riemann equations and "impact upon imaginary" are used in the form of Laplace's equation - showing even a simple single PDE is a touchy thing. It uses computer modeling to show possible directions involved in the parent equation of the partial differential, a backtrace analysis. I understand that modeling of 3space directions upon a surface of gradients (and thus possible anti-derivatives of integrals of these) can be done. I checked:, but it has no proof or methods than that of inspection. Solvers which are quite complicated code - and Wolfram does show this in the Help you can access: this post suggested by Neil helped me understand the overall "problem" with asking "what can NDSolve do and not do". You'd need a specific trace on a specific surface for an x,y,z solution: but PDE leave certain trace directions as "variable" and that's bound to trip up solvers ability to show all (relevant) sol'n. While I'm not much studied, I think you would be lucky if any of the PDE solvers gave you all solutions because PDE have many more degrees of freedom in solving than do ODE. ![]() though they solve many slab or wave equations. ![]() neither ODE book I have shows proofs that prove solutions to systems of PDE (which are more variate) being the same as systems of ODE (which can always be reduce in order). (2) (3) are a hypothesis of a "systems of PDE". conditions upon it reduce the outer equations solutions. (1) is a common (ie thermal / slab) PDE that any ODE book discusses. A1 = 1 A2 = 1 A3 = 1 A4 = 1 \ = 1 \ = 1 īi = 1 Subscript = 1 eq =, Mesh -> None, ColorFunction -> Hue, It is not possible to solve the problem analytically. ![]() OK, I translated into the Wolfram Language and made a code for numerical integration.
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