A NOVEL TEMPERED FRACTIONAL TRANSFORM: THEORY, PROPERTIES AND APPLICATIONS TO DIFFERENTIAL EQUATIONS

In this paper, we develop a new technique known as Tempered Fractional J-Transform (TFJT). This scheme can be applied to study numerous linear and nonlinear dynamical systems in tempered fractional (TF) calculus in both Riemann-Liouville and Caputo and sense. Some new theories, properties, and applications of the above-mentioned J-transform are calculated in detail. The proofs of some important theorems on TF Riemann-Liouville and Caputo derivatives are proved based on TFJT. For validation, accuracy and efficiency, the general TF equations as well as TF linear and nonlinear Klein-Gordon equations are studied by using the proposed transform with the numerical illustrations. It is observed that the proposed technique is fast convergent and the results are the first precise confirmations of TFJT in tempered calculus for nonlinear systems. This work can be studied as a substitute to present mathematical methods and will have extensive applications in physical sciences.