ELSAKI TRANSFORMATION AND ITS APPLICATION - Sagindykov, Б.Ж.; Bimurat, Ж. (Bulletin of Abai KazNPU. Series of Physical and Mathematical sciences, 2026)
The Elzaki transform is an integral transform analogous to the Laplace transform. It is a modified version of the Laplace transform and serves as a mathematical tool for analyzing engineering and physical problems. This paper explores the fundamental properties of the Elzaki transformation, which remains relatively unknown to Kazakh readers and is rarely applied in practice. Unlike conventional transforms, the Elzaki transform enables problem-solving without transitioning to a new frequency domain. In this transformation, a function of a real variable can correspond to either a real- or complex-valued function, referred to as its image. If the given function is real-valued, its Elzaki transform is generally a real-valued function as well. However, in certain cases, particularly when dealing with exponential and trigonometric functions, the image may also be complex-valued. Therefore, this study examines the key properties and theorems of the Elzaki transform to establish a set of image correspondences. As a result, we construct a table mapping original functions to their transformed counterparts. The primary objective of this paper is to demonstrate the application of the Elzaki transform in solving linear differential equations with constant and variable coefficients. Additionally, we present generalized shift theorems and analyze the transform’s effectiveness in differential equation solutions. Finally, a comparative analysis of the Laplace, Sumudu, and Elzaki transforms is conducted, highlighting their interrelationships.