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An innovative, extremely fast and accurate method is presented for Neumann-Dirichlet and Dirichlet-Neumann boundary problems for the Poisson equation, and the diffusion and wave equation in quasi-stationary regime; using the finite difference method, in one dimensional case. Two novels matrices are determined allowing a direct and exact formulation of the solution of the Poisson equation. Verification is also done considering an interesting potential problem and the sensibility is determined. This new method has an algorithm complexity of
O(
N), its truncation error goes like
O(
h
^{2}), and it is more precise and faster than the Thomas algorithm.

Poisson equation is used to describe, in quantitative manner, electrostatic and magnetostatic phenomena. It also helps to understand diffusion and propagation related problems, in quasi-stationary regime. Its solution is of great interest for a wide range of fields such as engineering, physics, mathematics, biology, chemistry, etc.

Most of solving methods, of this very important equation, use matrix inversion technics and algorithms, which are dependent on its Right-Hand Side (RHS). A recent study [

In the present study, we focus on the Poisson equation (1D), particularly in the two boundary problems: Neumann-Dirichlet (ND) and Dirichlet-Neumann (DN), using the Finite Difference Method (FDM). Essentially, attention is given to the matrices extracted from the algebraic equations from this differential method. Furthermore, an exact formulation of their inverses, independently of the RHS, is determined. Therefore, a new and advanced formulation of the solution to the Poisson equation, is found, for Neumann boundary conditions.

The proposed method is more accurate and faster than the Gaussian elimination method and that of Thomas. In addition, it completes the work made by Gueye S. Bira [

We will first consider an ND boundary problem and establish the corresponding algebraic equations coming from the application of the finite difference method, using the centered difference approximation (second order derivative). Then, we will, based on these algebraic equations, and considering the boundary conditions; establish the matrix equation. Thereafter, we discuss the properties of the associated matrix and then, determine its inverse, exactly and independently of the RHS. This will allow a direct and exact formulation of the solution to the Poisson equation for a 1D problem with ND boundary conditions. Complexity, accuracy, and stability are discussed and compared with other methods: Gaussian elimination algorithm and Thomas. Moreover, a verification of this new method is done by considering an interesting potential problem with inhomogeneous ND boundary conditions. The results are compared to the exact analytical solution and show great agreement. A similar approach is followed in the case Dirichlet-Neumann problem. The exact formula of the inverse matrix is determined and also the solution of the differential equation.

We consider a scalar potential

The mesh is composed of

point,

proximate value of the desired potential at point

and the second derivative:

Thus, the discretized 1D Poisson equation becomes a set of algebraic equations:

The boundary

One sees that this extra point does not affect the result. It is also to remark that the truncation error goes like

We can introduce the vector

Thus, one obtains the following matrix equation:

The centered difference approximation leads to an N × N-matrix

The inverse of the matrix

where

It also holds:

The behavior of the determinant and the co-factor of the matrix

Using the relations in (7)-(9), we can determine exactly the inverse of the matrix

Equation (10) is also equivalent to:

Equations (10) and (11) contain the same information. We prefer the first because it appears to be simpler than the latter and can be preferred for an eventual implementation in a programming language.

Thus, the inverse matrix is entirely determined. We get the simple, beautiful, exact, and very important matrix

We call this impressive matrix

The matrix

With Equation (6), one obtains the solution

The scalar potential

Thus, the solution of the 1D Poisson equation, in the case of Neumann-Dirichlet boundary, is determined exactly with the direct relation:

This is equivalent to:

Equation (15) represents a great improvement for solving the Poisson equation, particularly for Neumann- Dirichlet boundary conditions. The solution is determined properly, exactly, and given in a direct formulation. It can be very easily programmed. One loop will be largely sufficient to compute all the solution of one the most important equation in physics and engineering, in the one-dimensional case. It is a novel and exact formulation of the solution with the finite difference method using the centered difference approximation. The very important matrix

The methods that use inversion technics to obtained the matrix

The presented new solution is more direct, more exact, more stable; and faster than the Thomas Method for 1D Poisson equation. An important fact is that the determination of

We consider a scalar field

in

We can apply the finite difference method, taking:

Then, we compute the solution, with new method, given by Equation (15) and compare it with the exact potential (Equation (16)). Naturally, we also take into account the Equation (5).

We denote by

For a given

The denote

It is calculated for the given parameters and its value is:

the centered approximation. It also gives the exact value of the potential

We see that the solution of the ND boundary problem with the proposed method is also very accurate as shown in the table above.

At this stage, we are interested in the sensitivity of this method. We have shown the average relative error

tion can be assumed to be proportional to

A curve fitting of the sensibility can be given with:

where

The average relative error

For the given function

As we proceeded in the case of boundary conditions of type ND; we will do the same for a DN problem. The first step is to find an adequate and comfortable discretization. We propose that of

Here, the mesh points

Thus, the vector

Thus, the matrix equation becomes:

In the case of DN boundary conditions, the matrix

Thus, the inverse matrix of

We call this impressive matrix

This solution, given by the simple and extremely important Equation (24), can be easily computed, in one programming loop that give all the solutions.

We consider the same potential as that of the ND boundary problem, studied above. In this DN problem, the

boundary conditions are:

We can apply the finite difference method, taking:

Then, we compute the solution, of our new method, given by Equation (24) and compare it with the exact potential.

The solution of the DN problem is also very accurate as shown in

Now, the sensibility can be determined, for the DN boundary problem: the average relative error

A curve fitting of the sensibility can be given using Equation (20) with

The average relative error

This study has determined two novels matrices independently of the RHS providing a new and exact formulation of the solution of the Neumann boundary problem, for the 1D Poisson equation. The presented results and methods constitute a great improvement in the field of solving similar equations: diffusion and wave equations, in the quasi-stationary case, using the FDM. They are direct, highly accurate, extremely fast, and economical in terms of memory occupation.