Integral transform method plays a significant role to solve different kinds of integral and differential equations of fractional order and integer order that arise in several applications of engineering and physical sciences. Natural transform converges to Laplace and Sumudu transforms. The aim of this present paper is to solve the unified fractional Schrödinger equation that occurs in the field of quantum mechanics. The solution is obtained by Natural transform technique for the Caputo fractional derivatives with time variable and Fourier transform for the Liouville fractional derivative with space variable. The result is provided in the computational form of the H-function. The paper explicitly reveals the efficiency and reliability of joint Natural and Fourier transform technique. Some special cases of the main result are mentioned. Achieved result is general in nature and has the capability of providing a number of known and new results.
Special functions have enormous applications in theoretical and applied mathematics. This field is constantly expanding and addressing new problems in engineering applications and applied sciences. H-function one and more variables have been applied in large range of areas, such as electronics and communication, astrophysics, fractional differential equations, super statistics, diffusion, reaction–diffusion, and other fields like probability theory, biology, and theoretical physics. Some situations of heat conduction and astrophysics problems are not adequately addressed by basic category of special functions. To address such problems, incomplete special functions found the way in the literature. Incomplete H-functions are attracting researchers working on heat conduction and astrophysics problems. The paper aims to transform the products of algebraic powers and incomplete H-function in another domain using inverse laplace transform. Results are obtained in terms of the incomplete H-function as compact form. Further, some particular cases are mentioned by giving special values to the incomplete H-function’s parameters.
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