The Klein Krames equation (KKE) stands for the probability distribution function (PDF) that describes the diffusion of particles subjected an external force. It is shown that 2 A. A. Alderremy, H. I. Abdel-Gawad, Khaled M. Saad, Shaban Aly the conformable fractional derivative (CFD) KKE can be reduced to the classical one's by using similarity transformations. Here, the objective of this work is to find the exact solutions of CFD-KKE.. To this issue, an approach is presented. It is based on transforming the KKE to a system of first order PDEs. The solutions are found by implementing extended unified method. It is found that, the integrability condition is that the external force is constant. The numerical results of the solutions are calculated and the are shown graphically. 2020 Mathematics Subject Classification: 34A08; 35A22; 41A30; 65N22.

This article analyzes and compares the two algorithms for the numerical solutions of the fractional isothermal chemical equations (FICEs) based on mass action kinetics for autocatalytic feedback, involving the conversion of a reactant in the Liouville-Caputo sense. The first method is based upon the spectral collocation method (SCM), where the properties of Legendre polynomials are utilized to reduce the FICEs to a set of algebraic equations. We then use the well-known method like Newton-Raphson method (NRM) to solve the set of algebraic equations. The second method is based upon the properties of Newton polynomial interpolation (NPI) and the fundamental theorem of fractional calculus. We utilize these methods to construct the numerical solutions of the FICEs. The accuracy and effectiveness of these methods is satisfied graphically by combining the numerical results and plotting the absolute error. Also, the absolute errors are tabulated, and a good agreement found in all cases. 2020 Mathematics Subject Classification: 34A08; 35A22; 41A30; 65N22.

In this paper we utilize the new operators using three different kernels,namely, power law, exponential decay, and the generalized Mittag-Leffler function based on the fractal-fractional differentiation. These constructed operators have two parameters: the first is considered as fractional order and the second as fractal dimension. We studied the effect of the fractal-fractional derivative order as well as the nonlinear term order 1 < q ≤ 2 on the behavior of the numerical solutions of the fractal-fractional reaction diffusion equations (FFRDE). The iterative approximations are constructed by applying the theory of fractional calculus with the help of Lagrange polynomial functions. In the case of β = k = 1 , all the numerical solutions based on power kernel, exponential kernel, and the generalized Mittag-Leffler kernel are very close to each other and so one of the kernel was compared with numerical methods like finite difference methods (FDM) and an excellent agreement obtained. All calculations in this paper were done using the mathematica package.

This article investigates a family of approximate solutions for the fractional isothermal chemical (FIC) equation based on mass action kinetics for autocatalytic feedback, involving the conversion of a reactant in the Liouville-Caputo sense. We apply two methods to construct numerical solutions of the FIC equation. By the first method, the spectral colloca-tion method (SCM), we reduce the FIC equation to a system of algebraic equations using Chebyshev polynomials of the third kind (CPTK). We then use the Newton-Raphson method (NRM) to solve the system of algebraic equations. By the second method, using properties of Lagrange polynomial interpolation (LPI) after applying the fundamental theorem of fractional calculus, we evaluate numerical solutions of the FIC equation. We compare these numerical solutions and compute the absolute error for varying parameter values. The results confirm the efficiency of the methods and their computationally favorable use for the numerical treatment of the model equations. 2020 Mathematics Subject Classification: 34A08; 35A22; 41A30; 65N22.