Numerical simulation of telegraph and Cattaneo fractional‐type models using adaptive reproducing kernel framework

Abstract

In this article, a class of generalized telegraph and Cattaneo time-fractional models along with Robin's initial-boundary conditions is considered using the adaptive reproducing kernel framework. Accordingly, a relatively novel numerical treatment is introduced to investigate and interpret approximate solutions to telegraph and Cattaneo models of time-fractional derivatives in Caputo sense. This treatment optimized solutions relying on the Sobolev spaces and Schmidt orthogonalization process that can be directly implemented to generate Fourier expansion at a rapid convergence rate, in which the arbitrary kernel functions satisfy Robin's homogeneous conditions. Furthermore, the solution is displayed in a fractional series formula in complete Hilbert spaces without any restrictive hypothesis on the desired issues. The effectiveness, validity, and potentiality of the proposed procedure are demonstrated by testing some applications. The graphical consequences indicate that the method is superior, accurate, and convenient in solving such fractional models. K E Y W O R D Sfractional Cattaneo equation, fractional partial differential equations, fractional telegraph equation, reproducing kernel method, Robin's initial-boundary conditions | INTRODUCTIONMathematical and computational modeling utilizing partial differential equations (DEs) (PDEs), in a fractional sense, better describes the complexity of certain factual structures and facilitates the understanding of physical dynamic processes in terms of spatial and temporal parameters dominated by various external forces that influence behavior to become more sophisticated and unpredictable. [1][2][3] However, at a pilot level, this process requires extensive empirical analysis going through several stages, the essence of which is a thorough examination of the past and current situation, including relevant information, effects, and sources, then building design formulation based on mathematical and logical relationships and so validating and developing the model created, while a model solution and interpretation of the results obtained is not a stage of the modeling process itself.On the other aspect as well, the numerical investigation of fractional PDEs has grown rapidly and has gained more attention over the past few decades due mainly to its elegant role in modeling emerging realism issues in various scientific fields. Indeed, it provides an excellent instrument to formulate many nonlinear problems that occur in applied sciences with a typical set of time and space fractional derivatives involving nice features for various materials specific to hereditary and memory as well as to simplify the control design without altering dynamic modeling structure. Recently, numerous applications of fractional DEs can be found in chemistry, biology, electrochemistry,

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