Numerical Computation and Exact Solutions for Stokes Flow Problems Involving Non-uniform Electric Fields

The biharmonic equation is a mathematical problem that arises in various fields such as physics, engineering, and computer science. It is a second-order differential equation that describes the behavior of physical phenomena such as lightning or water flow. However, solving this equation can be challenging due to its nonlinear nature. In this article, we propose a new method for solving the biharmonic equation using rational approximation.
Rational approximation is a mathematical technique that involves approximating a complex function with a simpler one, called a rational function. This technique has been widely used in various fields, including signal processing and image analysis. By using rational approximation, we can simplify the biharmonic equation into a linear system of equations, which can be easily solved.
The key idea behind our method is to construct a rational function basis that spans the same spaces as the original basis. This means that we create a new set of basis functions that are similar to the original ones but are more convenient to work with. Once we have this basis, we can use it to approximate the solution of the biharmonic equation.
To construct the rational function basis, we perform VA orthogonalization for the Newman terms, the Runge terms, and the Laurent series corresponding to each hole for all sample points. This gives us a well-conditioned rational function basis that allows us to evaluate every sample point in the domain and its derivative.
The advantages of our method are several. Firstly, it is fast and efficient, allowing us to solve the biharmonic equation quickly and accurately. Secondly, it provides a more straightforward way of solving the equation, which can be useful for complex problems. Finally, it allows us to obtain a more accurate approximation of the solution by including more terms in the rational approximation.
In conclusion, our method offers a new and effective way of solving the biharmonic equation using rational approximation. By constructing a rational function basis that spans the same spaces as the original basis, we can simplify the equation and solve it quickly and accurately. This technique has many applications in various fields, including physics, engineering, and computer science.