Solving Linear Ordinary Differential Equations with Variable Coefficients Using a New Integral Transformation
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The new integral transform, Linear ordinary differential equations, Variable Coefficients, Initial value problem, Inverse of the New Integral Transform
Abstract
This research aims to introduce and apply a new integral transformation for solving linear ordinary differential equations with variable coefficients, such equations are considered among the most complex mathematical models, posing significant challenges in various areas of applied mathematics. In this context, a new integral transformation is proposed, based on a specially constructed kernel function that facilitates the handling of non-constant coefficients. The transformation is rigorously defined, and the necessary conditions for its existence are established. Its fundamental properties are then examined, including its application to basic functions such as polynomials and exponential functions. Furthermore, the inverse transformation is derived, allowing the recovery of the original function upon completion of the transformation process. The transformation rules for first and second order derivatives are also obtained, with indication of how these rules can be generalized to derivatives of higher orders. The efficiency of this transformation is evaluated by applying it to a set of initial value problems for differential equations. The results demonstrate that the new integral transformation provides accurate analytical solutions without requiring complex mathematical manipulations or reliance on approximate numerical methods. A comparative analysis between the solutions obtained using this transformation and those derived via the Laplace transformation reveals that the proposed method may exhibit superiority in certain cases in terms of accuracy and simplicity. The findings of this study suggest that the new integral transformation can serve as a powerful mathematical tool for analyzing differential equations with variable coefficients and may pave the way for its application to more complex systems of differential equations and partial differential equations. As such, this transformation represents an innovative contribution to the field of mathematical transformations, broaden their future applications and enhancing their role as effective tools for solving differential equations. Additionally, a set of MATLAB scripts has been developed to automate the computation of the new integral transformation for functions of this form t^n f(x),t^n f^' (x),t^n f''(t), where n is a positive integer. These computational tools have significantly accelerated the solution process and reduced the computational burden, thereby reinforcing the potential of this transformation for practical and engineering applications.
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