Ordinary differential equations or (ODE) are equations where the derivatives are taken with respect to only one variable. That is, there is only one independent variable. Partial differential equations or (PDE) are equations that depend on partial derivatives of several variables.
When you solve some equation you are trying to get the values that satisfy those equations. But when you are solving ODE/PDE Equations you are trying to get the entire definition of the "Function" that satisfies those ODE/PDE Equations. That is why the latter is extremely difficult to solve and is very important to Physics, Engineering and Sciences. Because every system in this Universe literally works on and can be described by ODE/PDE Equations.
Partial Differential Equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, thermodynamics, etc. literally everything we know.
Solving ODE/PDEs normally requires HPC compute infrastructure and takes a lot of time.
Automatski' solvers can solve them in near Realtime and deliver an Exponential Computational Advantage in ODE PDE Solutions.
The problem remains - what is the most generic way of defining such problems so that our solvers can then solve them?
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