# Solving partial differential equation numerically

• Telemachus
In summary, the author set up a problem to solve using first order partial differential equations. However, the matrix is singular and the correct approach is not currently known.
Telemachus
Hi there. I am trying to self teach how to solve partial differential equations numerically using finite differences. I know this is a complex field, that requires much more knowledge of the theory than what I actually know, but anyway I wanted to try.

Anyway, I've tried to build my own differential equation, probably naively. I wanted to solve a first order partial differential equation, so I thought of taking the function:

##f(x,y)=sin(x-y)##

And then, taking derivatives I build the differential equation:

##\displaystyle \frac{\partial f(x,y)}{\partial x}+\frac{\partial f(x,y)}{\partial y}=0##, and I considered the interval ##0\leq x \leq 1, 0\leq y \leq 1##

Then I considered the boundary conditions ##f(0,y)=\sin(-y)##, ##f(1,y)=\sin(1-y)##,##f(x,0)=\sin(x)## and ##f(x,1)=\sin(x-1)##.I think that everything is correct until here, but I'm not sure, if anything is wrong please tell me.

Then I considered centered differences (I needed for centered difference to have the previous and forward steps in the equation, in order to propagate the solution from the boundaries). I've considered equal spacings in the y and x directions, with ##x_i=i\Delta x=ih##, ##y_j=j\Delta y=j h##, and ##h=\frac{1}{N+1}##, with N being the number of interior points in each axis, being ##(x_0,y_j),(x_{N+1},y_j)## and ##(x_i,y_0),(x_i,y_{N+1})## the points situated at the boundaries of the domain.

Under this conditions I've arrived to the set of equations:

##u_{i+1,j}-u_{i-1,j}+u_{i,j+1}-u_{i,j-1}=0##

Where I thought that ##u_{i,j}\sim f(x_i,y_j)##

Then the discretized version of the boundary conditions would look like: ##u_{0,j}=\sin(-jh)##, ##u_{N+1,j}=\sin(1-jh)##, ##u_{i,0}=\sin(ih)## and ##u_{i,N+1}=\sin(ih-1)##.

But when I build the matrix and solve the linear system I get the result that the matrix is singular. So what would be the correct approach to this problem?

Yes, but those are second order. I was interested in solving a first order partial differential equation. Is anything wrong with the problem I set up? is it just that the numerical approach isn't proper for this equation or I did something else wrong?

## What is a partial differential equation (PDE)?

A partial differential equation is a mathematical equation that involves multiple independent variables and their partial derivatives. It is used to model physical phenomena in fields such as physics, engineering, and economics.

## Why do we need to solve PDEs numerically?

Many PDEs cannot be solved analytically, meaning there is no exact solution that can be written in a closed form. Therefore, numerical methods are used to approximate the solution and provide useful insights into the behavior of the system.

## What are some common numerical methods for solving PDEs?

Some common numerical methods for solving PDEs include finite difference methods, finite volume methods, and finite element methods. These methods discretize the PDE into a set of algebraic equations that can be solved using computer algorithms.

## What are the advantages of solving PDEs numerically?

Numerical methods allow for the solution of complex PDEs that cannot be solved analytically. They also provide a more efficient and accurate approach compared to analytical methods, especially for high-dimensional problems.

## What are some challenges in solving PDEs numerically?

One of the main challenges in solving PDEs numerically is the trade-off between accuracy and computational cost. As the grid size increases, the accuracy of the solution improves, but the computational cost also increases. Another challenge is ensuring numerical stability, as some methods may produce unstable solutions if not implemented properly.

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