Solution of a non-homogeneous dynamic equation on a time scale

Azaba, Ahmed; Raslan, Kamal R.; Ali, Khalid K.

doi:10.22034/cmde.2025.65428.3009

Solution of a non-homogeneous dynamic equation on a time scale

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.

³ Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman.

10.22034/cmde.2025.65428.3009

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 the 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

Approximation methods and numerical treatment of dynamical systems

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