# Computer-Based Solving of Partial Differential Equations Using the Method of Nets

## Description

This paper explores numerical methods for solving partial differential equations (PDEs) using the method of nets. The focus is on hyperbolic equations, such as the wave equation, and the application of net methods in solving problems with boundary conditions. The process of solving these equations using computational tools is illustrated, and the accuracy of the results is analyzed. The iterative Gauss-Seidel method is applied to solve systems of algebraic equations generated by the net method

## Files

## Steps to reproduce

Steps to Reproduce the Method of Nets: Problem Setup: Select a PDE: Choose the specific partial differential equation you want to solve, such as the wave equation, heat equation, or Poisson equation. Define Conditions: Establish the initial conditions (values of the function at the starting time) and boundary conditions (values of the function at the spatial boundaries) relevant to your problem. Discretize the Domain: Create a Grid: Divide the spatial and temporal domains into discrete points to form a computational grid or mesh. Label Grid Points: Assign indices to each grid point to represent their positions in space and time, typically using two indices—one for space and one for time. Discretize the Equation: Approximate Derivatives: Replace the continuous partial derivatives in the PDE with finite difference approximations using values at the grid points. Formulate Algebraic Equations: This substitution transforms the PDE into a system of algebraic equations that relate the function values at different grid points. Construct the System of Equations: Combine Equations: Assemble the finite difference approximations into a comprehensive system of algebraic equations, ensuring consistency across the grid. Ensure Correspondence: Each algebraic equation should correspond to a specific grid point in your computational domain. Apply Boundary and Initial Conditions: Implement Initial Conditions: Set the function values at the initial time based on the problem's initial conditions. Implement Boundary Conditions: Apply the boundary conditions by setting the function values at the spatial boundaries for all time steps. Solve Iteratively Using the Gauss-Seidel Method: Initial Guess: Start with an initial approximation for the function values at all grid points. Iterative Update: Use the Gauss-Seidel method to iteratively update the function values. This involves sequentially solving for each variable while using the most recent updates. Convergence Check: Continue the iterations until the changes between successive updates are smaller than a predetermined tolerance level, indicating that the solution has stabilized. Visualize the Results: Data Presentation: After obtaining the numerical solution, organize the data for analysis. Graphical Representation: Use software tools (such as MATLAB or Python with Matplotlib) to create visual representations like plots or surface graphs that illustrate how the function behaves over space and time. Compare with Analytical Solution (If Available): Analytical Benchmark: If an exact analytical solution exists for the PDE, calculate it for the same conditions. Error Analysis: Compare the numerical results with the analytical solution to evaluate the accuracy of the numerical method. Assessment: Analyze any discrepancies to understand the limitations or errors introduced by the discretization and numerical methods.