Solving general PDE’s in SymPy

Hello, I have decided to start my official work for my GSoC project. This post would be slightly TL;DR , so please bear with me.

1. The first step was to wrap up the work on my previous PR, I had added a hint such that SymPy can now solve general partial differential equations with constant coefficients.  The mentioned Partial Differential Equation is of the form \frac{\partial f}{\partial x} + b*\frac{\partial f}{\partial y} + c*f =G(x,y) where f is a function of x and y and G can be any arbitrary function in x and y.

The implementation is discussed as follows, it can be separated into a general part and homogeneous part. The homogeneous part of course is  a*\frac{\partial f}{\partial x} + b*\frac{\partial f}{\partial y} + c*f and the general part is when you add the function G(x,y) and the solution would be when you add the general part and homogeneous part.

So how does one solve the homogeneous part. It beats me too. I had referred this book “Partial Differential Equations by Viktor Grigoryan” and this is what he says in the book.  Before I go into the details of the implementation, it is important to review a few concepts that I had studied in my Math-1 Course in college.

a] Directional Derivatives –  All of us know that a derivative is the rate of change of  a function. \frac{\partial f}{\partial x} is the rate of change in the x direction, similarly \frac{\partial f}{\partial y}  is the rate of change in the y direction. Suppose one wants to find the rate of change in an arbitrary direction say , this is the way to proceed. We find the dot product between the unit vector which gives the direction , and the gradient of the function, which would give the rate of change of function in the direction \frac{v}{|v|}.(\frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j}) . The reasoning behind it? Well I’ll leave it to someday else.

Solving the homogeneous part : Now take the Partial Differential Equation a*\frac{\partial f}{\partial x} + b*\frac{\partial f}{\partial y} + c*f . Our goal is to convert it this PDE in two variables into an ordinary differential equation, the first additive part can be written as directional derivative of f in the direction of a\hat{i} + b\hat{j} . It can be also seen the third part c*f(x,y) is independent of the derivative term. Just ignoring the third term for a moment, the first part becomes zero, which tells us that it would be convenient to change the co-ordinates from the x and y to a system parallel to a\hat{i} + b\hat{j} and perpendicular to it. So we make the following substitutions

\xi=a*x+by and \eta=b*x-a*y and yay the homogeneous PDE reduces to, (a^{2} + b^{2})*\frac{\partial u}{\partial \xi} + u=0 . Do the substitutions yourself. It isn’t that difficult, and the ODE is variable separable and the solution is given by e^{-c/(a^{2} + b^{2})*(a*x + b*y)}*f(b*x - a*y) . This solves the homogeneous part of the Partial Differential Equation

Solving the general part : The general Partial Differential Equation reduces to the form \frac{\partial u}{\partial \xi} +\frac{c*u}{a^{2} + b^{2}}= \frac{G(\xi , \eta)}{a^{2} + b^{2}} , which is a linear differential equation. The solution is given by,  e^{-c/(a^{2} + b^{2})*(a*x + b*y)}*\int\frac{e^\frac{c*\xi}{a^{2} + b^{2}}*G(\xi , \eta)}{a^{2} + b^{2}}\,d\xi.

The corresponding source code can be found here, for those who are interested

I’ve also finished work on implementing the first heuristic, however I did not want to squeeze it here, as the post is already going too big, I have to admit that the code was completely straightforward and formatting this post was a tougher task. So till next time.

P.S : Going through emotional lows now which I wouldn’t want to mention here , just hoping tomorrow is better than today.

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