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This article is devoted to both theoretical and numerical studies of eigenvalues of regular fractional 2α-order Sturm-Liouville problem where 1/2 < α < 1. In this paper, we implement the modified fractional power series (MFPS) method to approximate the eigenvalues. To find the eigenvalues, we force the approximate solution produced by the MMPS method satisfies the boundary condition at x = 1. The fractional derivative is described in the Caputo sense. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the eigenfunctions of the proposed problem. The convergence of the approximate eigenfunctions produced by the MFPS to the exact eigenfunctions is proven. [ABSTRACT FROM AUTHOR]

In this paper, we study the pantograph equation. The reproducing kernel Hilbert space method (RKHSM) is employed to compute an approximation to the solution of this problem. The validity of the RKHSM is ascertained by comparing our results with other methods in the literature. The results reveal that the proposed analytical method can achieve excellent results in predicting the solutions of such problems. The existences of the solution is proved. In addition, the uniform convergent of the proposed method is investigated. [ABSTRACT FROM AUTHOR]

In this work, a functional operator extracted from Korsunsky's technique is used to produce new two-mode nonlinear equations. These new equations describe the motion of two directional solitary-waves overlapping with an increasing phase-velocity and affected by two factors labeled as the dispersion and nonlinearity coefficients. To investigate the dynamics of this two-mode family, we construct the two-mode KdV-Burgers-Kuramoto equation (TMKBK) and two-mode Hirota-Satsuma model (TMHS). Two efficient schemes are used to assign the necessary constraints for existence of solutions and to extract them. The role of the phase-velocity on the motion of the obtained two-wave solutions is investigated graphically. Finally, all the obtained solutions are categorized according to their physical shapes., (© 2020 The Author(s).)

The purpose of this work is to analytically simulate the mutual impact for the existence of both temporal and spatial Caputo fractional derivative parameters in higher-dimensional physical models. For this purpose, we employ the γ̅ -Maclaurin series along with an amendment of the power series technique. To supplement our idea, we present the necessary convergence analysis regarding the γ̅ -Maclaurin series. As for the application side, we solved versions of the higher-dimensional heat and wave models with spatial and temporal Caputo fractional derivatives in terms of a rapidly convergent γ̅ -Maclaurin series. The method performed extremely well, and the projections of the obtained solutions into the integer space are compatible with solutions available in the literature. Finally, the graphical analysis showed a possibility that the Caputo fractional derivatives reflect some memory characteristics., Competing Interests: Competing interestsThe authors declare that there are no conflicts of interest regarding the publication of this article., (© The Author(s) 2020.)

In this paper, we investigate the simultaneous approximation of a function f ( x ) and its derivative [Formula: see text] by Hermite interpolation operator [Formula: see text] based on Chevyshev polynomials. We also establish general theorem on extreme points for Hermite interpolation operator. Some results are considered to be an improvement over those obtained in Al-Khaled and Khalil (Numer Funct Anal Optim 21(5-6): 579-588, 2000), while others agrees with Pottinger's results (Pottinger in Z Agnew Math Mech 56: T310-T311, 1976).