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    Applications of generalized formable transform with ?-Hilfer-Prabhakar derivatives
    (Springer Heidelberg, 2024) Khalid, Mohd; Mallah, Ishfaq Ahmad; Akgul, Ali; Alha, Subhash; Sakar, Necibullah
    This paper introduces the Psi-formable integral transform, discusses the several essential properties and results-Convolution, Psi-formable transform of tth derivative, Psi-Riemann Liouville fractional integration and differentiation, Psi-Caputo fractional differentiation, Psi-Hilfer fractional differentiation, Psi-Prabhakar fractional integration and differentiation, and Psi-Hilfer-Prabhakar fractional derivatives. Next, we use the Fourier integral and Psi-Modifiable conversions to solve some Cauchy-type fractional differential equations using the generalized three-parameter Mittag-Leffler function and Psi-Hilfer-Prabhakar fractional derivatives
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    Exploring the Elzaki Transform: Unveiling Solutions to Reaction-Diffu-sion Equations with Generalized Composite Fractional Derivatives
    (Universal Wiser Publisher, 2024) Khalid, Mohd; Mallah, Ishfaq Ahmad; Alha, Subhash; Akguel, Ali
    This article investigates the use of the Elzaki transform on a generalized composite fractional derivative. To establish the framework for this inquiry, numerous essential lemmas about the Elzaki transform are presented. We successfully extract the solution to the reaction-diffusion problem using both the Elzaki and Fourier transforms, which include a generalized composite fractional derivative. We also look at special examples of the generalized equation, which helps us understand its applications and consequences better. The results show that the Elzaki transform is successful in dealing with complicated fractional differential equations, introducing new analytical approaches and solutions to the subject of fractional calculus and its applications in reaction-diffusion systems.
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    New Aspects of Bloch Model Associated with Fractal Fractional Derivatives
    (De Gruyter Open Ltd, 2021) Akgül, Ali; Mallah, Ishfaq Ahmad; Alha, Subhash
    To model complex real world problems, the novel concept of non-local fractal-fractional differential and integral operators with two orders (fractional order and fractal dimension) have been used as mathematical tools in contrast to classical derivatives and integrals. In this paper, we consider Bloch equations with fractal-fractional derivatives. We find the general solutions for components of magnetization M = (Mu, Mv, Mw) by using descritization and Lagrange's two step polynomial interpolation. We analyze the model with three different kernels namely power function, exponential decay function and Mittag-Leffler type function. We provide graphical behaviour of magnetization components M = (Mu, Mv, Mw) on different orders. The examination of Bloch equations with fractal-fractional derivatives show new aspects of Bloch equations. © 2021 Ali Akgül et al., published by De Gruyter.
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    On ?-Hilfer generalized proportional fractional operators
    (Amer Inst Mathematical Sciences-Aims, 2021) Mallah, Ishfaq; Ahmed, Idris; Akgul, Ali; Jarad, Fahd; Alha, Subhash
    In this paper, we introduce a generalized fractional operator in the setting of Hilfer fractional derivatives, the psi-Hilfer generalized proportional fractional derivative of a function with respect to another function. The proposed operator can be viewed as an interpolator between the Riemann-Liouville and Caputo generalized proportional fractional operators. The properties of the proposed operator are established under some classical and standard assumptions. As an application, we formulate a nonlinear fractional differential equation with a nonlocal initial condition and investigate its equivalence with Volterra integral equations, existence, and uniqueness of solutions. Finally, illustrative examples are given to demonstrate the theoretical results.