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In this article, the authors introduce the [Formula: see text]-transform and derive the complex inversion formula and a convolution theorem for the transform. Furthermore, the fundamental solution of the Cauchy type fractional diffusion equation on fractals is given by means of the [Formula: see text]-transform in terms of the Wright functions. Also, the Cauchy fractional disturbance equation with continuous or discrete distribution of time fractional derivative is introduced and by using the [Formula: see text]-transform, its solution is expressed in terms of the Laplace type integral.

In this article,we derive a complex inversion formula and some new theorems related to 2-transform defined in [1],[2],[3].We also give an application for solution to non-homogeneous wave equation.

In this paper, it has been shown that the combined use of exponential operators and special functions provides a powerful tool to solve certain class of generalized space fractional Laguerre heat equation. It is shown that exponential operators are powerful and effective method for solving certain singular integral equations and space fractional Black–Scholes equation.

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Due to the need and the necessity to express a physical phenomenon in terms of an effective and comprehensive analytical form, this paper is devoted to study of Airy functions, which arise from the Airy differential equations, by means of integral transforms. Illustrative examples are also provided. The result reveals that the integral transforms are very useful tools to solve differential equations.