Solution of non-homogeneous dynamic equation on time scale

Articles in Press

Document Type : Research Paper

Authors

1 Al-Ryada University for Science and Technology, Sadat City, Menoufia, Egypt.

2 Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt.

Abstract

We find the general solution to the non-homogeneous dynamic equation, which is a combination of discrete and continuous mathematics. The nature of this equation requires a careful approach, as it involves elements of both types of math, making it both versatile and challenging. To address this, we will derive the formula for the general solution of the non-homogeneous equation, incorporating given initial conditions. During this process, we will define several critical points that may be either discrete, continuous, or a mix of both. By analyzing these points, we aim to capture the essence of the dynamic behavior of the system. Our approach involves finding an analytical solution to the equation and comparing it with a numerical approximation to evaluate their accuracy. We will graph both the analytical and numerical solutions to visualize their behavior and identify any discrepancies. Additionally, we will calculate the absolute error between the exact solution and the numerical solution to quantify the differences precisely. This comparison provides valuable insights into the accuracy and stability of numerical methods for solving such equations. Finally, we will demonstrate this approach by applying it to various examples, showcasing the methodology's effectiveness in solving a range of non-homogeneous dynamic equations with different initial conditions and parameters.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 11 May 2025

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History

  • Receive Date: 12 January 2025
  • Revise Date: 02 May 2025
  • Accept Date: 10 May 2025

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