# General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

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## Abstract

**:**

## 1. Introduction

## 2. The Exact Schemes

#### 2.1. The Concept of Local Green Functions

**Theorem**

**1**

**.**Consider the $[{x}_{i},{x}_{i+1}]$ subinterval of the partition (2) of interval $[0,\ell ]$. Let ${\psi}_{i}\left(x\right)$ and ${\phi}_{i}\left(x\right)$ be the test functions defined on subinterval $[{x}_{i},{x}_{i+1}]$ based on (3) and (4). The test functions ${\psi}_{i}\left(x\right)$ and ${\phi}_{i}\left(x\right)$ are equal at the appropriate endpoints of the $[{x}_{i},{x}_{i+1}]$ interval, i.e.,:

**Proof**

#### 2.2. Dirichlet Boundaries

**Theorem**

**2**

**.**If ${\psi}_{i-1}\left(x\right)$ and ${\phi}_{i}\left(x\right)$ are the test functions [9] satisfying the Equation (3), the Equation (4), respectively, and Theorem 1, the solution vector from following the exact scheme yields to the values of the analytical solution of Equation (1) at the grid points with boundary conditions in Equation (9). Equations (3) and (4) are special IVPs defined based on the homogeneous equation corresponding to the original ODE (Equation (1)).

**Proof**

#### 2.3. Exact Scheme for Dirichlet and Neumann Boundaries

## 3. Case Studies with Discontinuous $\mathit{\kappa}\left(\mathit{x}\right)$ and Singular ODE

#### 3.1. Implementation of the Exact Scheme to Solve Case Studies

- 1.
- After defining the ODE to be solved and the necessary boundary conditions (Dirichlet problem, mixed problem), based on Equation (2), we perform an arbitrary partition of the interval $[0,\ell ]$. The indexes of the nodes are: $i=0,1,\cdots ,n-1,n,n+1$, where 0 and $n+1$ are the indices of the boundary points.
- 2.
- 3.
- 4.
- Boundary conditions are also easily managed:
- For Dirichlet boundary conditions, we use the substitution defined in Equation (9),
- For mixed boundary conditions, we apply the Neumann to Dirichlet transformation based on the Equation (25) in order to calculate the potential value at the boundary point where the Neumann boundary condition is defined.

- 5.
- Construct the linear system of equations defined in Equation (10), using the coefficient values and boundary conditions calculated before, and solve it by inverting the tridiagonal matrix.
- 6.
- The elements of the vector obtained in this way are equal to the values of the analytic solution of the ODE defined in the first step taken in the grid points, according to the boundary conditions (according to Theorem 2).

#### 3.2. A Case Study for Discontinuous $\kappa \left(x\right)$ with Increasing Mesh Resolution

- 1.
- $n+1=16$ subintervals and the analytic solution is calculated in $n=15$ points,
- 2.
- $n+1=51$ subintervals and the analytic solution is calculated in $n=50$ points,
- 3.
- $n+1=101$ subintervals and the analytic solution is calculated in $n=100$ points.

#### 3.3. Case Study for a Singular ODE with a Piecewise Source Term

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Graphs of local green functions ${\psi}_{0}\left(x\right)$, ${\psi}_{1}\left(x\right)$, ${\psi}_{2}\left(x\right)$, and ${\psi}_{3}\left(x\right)$ (with solid lines) and ${\phi}_{1}\left(x\right)$, ${\phi}_{2}\left(x\right)$, ${\phi}_{3}\left(x\right)$, and ${\phi}_{4}\left(x\right)$ (with dotted lines) (defined in the Equation (31) using the partitioning in Equation (30)) on the interval $[0,2]$ ($n=3$).

**Figure 2.**The results of the case study: $u\left(x\right)$ is denoted with blue line, while ${u}_{i}$, the result obtained in the ${i}^{th}$ grid point, is denoted with black dots.

**Figure 4.**Graph of local green functions ${\psi}_{0}\left(t\right)$ and ${\phi}_{1}\left(t\right)$ if $a=1$, $b=2$ and $m=2$.

**Figure 5.**The solution ${u}_{0}\left(x\right)$ (in Equation (44)) of the case study for a singular ODE with a piecewise source term ($m=1,2,3,4$).

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**MDPI and ACS Style**

Vizvari, Z.; Klincsik, M.; Odry, P.; Tadic, V.; Sari, Z.
General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions. *Axioms* **2023**, *12*, 633.
https://doi.org/10.3390/axioms12070633

**AMA Style**

Vizvari Z, Klincsik M, Odry P, Tadic V, Sari Z.
General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions. *Axioms*. 2023; 12(7):633.
https://doi.org/10.3390/axioms12070633

**Chicago/Turabian Style**

Vizvari, Zoltan, Mihaly Klincsik, Peter Odry, Vladimir Tadic, and Zoltan Sari.
2023. "General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions" *Axioms* 12, no. 7: 633.
https://doi.org/10.3390/axioms12070633