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In this paper, we propose a fractional formulation, in terms of the Caputo derivative, of the Blasius flow described by a non-linear two-point fractional boundary value problem on a semi-infinite interval. We develop a finite difference method on quasi-uniform grids, based on a suitable modification of the classical L1 approximation formula and show the consistency, the stability and the convergence. The numerical results confirm the theoretical ones. Comparisons with some recently proposed results are carried out to validate the accuracy of the obtained numerical results, and to show the efficiency and the reliability of the proposed numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a new fractional derivative operator in which the generalized cardinal sine function is used as a non-singular analytic kernel. In addition, we provided the corresponding fractional integral operator. We expressed the new fractional derivative and integral operators as sums in terms of the Riemann–Liouville fractional integral operator. Next, we introduced an efficient extension of the new fractional operator that includes integrable singular kernel to overcome the initialization problem for related differential equations. We also proposed a numerical approach for the numerical simulation of IVPs incorporating the proposed extended fractional derivatives. The proposed fractional operators, the developed relations and the presented numerical method are expected to be employed in the field of fractional calculus. [ABSTRACT FROM AUTHOR]

This paper introduces a deterministic fractional-order epidemic model (FOEM) for studying the transmission dynamics of the human immunodeficiency virus (HIV) and acquired immunodeficiency syndrome (AIDS). The model highlights the substantial role of unaware and undetected HIV-infected individuals in spreading the disease. Control strategies, such as wielding condoms, level of preventive measures to avoid infection, and self-strictness of susceptibles in sexual contact, have been incorporated into the study. The basic reproduction number ℛ 0 α has been derived, which suggests the conditions for ensuring the persistence and elimination of the disease. Further, to validate the model, actual HIV data taken from Mexico and India separately have been used. The disease dynamics and its control in both countries are analyzed broadly. The values of biological parameters are estimated at which numerical solutions better match the actual data of HIV patients in the case of fractional-order (FO) instead of integer-order (IO). Moreover, in the light of ℛ 0 α , our findings forecast that the disease will abide in the population in Mexico, and at the same time, it will die out from India after a long time. [ABSTRACT FROM AUTHOR]

In this paper, a special class of fractional Sturm–Liouville problems of order α (3 < α ≤ 4) is discussed. The method is based on utilizing the shifted Legendre polynomials on the interval [ 0 , L ] to find numerical approximations for eigenvalues and corresponding eigenfunctions. The proposed method is implemented on several numerical examples. The numerical results indicate the high performance of the method, its effectiveness, and its simplicity in use. [ABSTRACT FROM AUTHOR]

This paper presents the study of fractional Emden–Fowler (FEF) equations by utilizinga new adequate procedure, specifically the q- homotopy analysis transform method (q- HATM). The EF equation has got greater significance in both physical and mathematical investigation of capillary and nonlinear dispersive gravity waves. The projected technique is tested by considering four illustrations of the time-fractional EF equations. The q -HATM furnish ℏ , known as an auxiliary parameter, by the support of ℏ we can modulate the various stages of convergence of the series solution. Additionally, to certify the resolution and accurateness of the proposed method we fitted the suitable numerical simulations. The redeem results guarantee that the proposed process is more convincing and scrutinizes the extremely nonlinear issues emerging in the field of science and engineering. • q -HATM is applied to obtain the solution to fractional Emden-Fowler equations. • Accuracy of the proposed method is validated through suitable numerical simulations. • The proposed method is efficient to study nonlinear issues in the areas of science. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized time fractional Burgers equation, where the time fractional derivative is in the sense of Caputo with derivative order in (0 , 1). If its solution u (x , t) has strong regularity, for example u (⋅ , t) ∈ C 2 [ 0 , T ] for a given time T , then we use the L1 scheme on uniform meshes to approximate the Caputo time-fractional derivative, and use the local discontinuous Galerkin (LDG) method to approach the space derivative. However, the solution u (x , t) likely behaves a certain regularity at the starting time, i.e., ∂ u ∂ t and ∂ 2 u ∂ 2 t can blow up as t → 0 + albeit u (⋅ , t) ∈ C [ 0 , T ] for a given time T. In this case, we use the L1 scheme on non-uniform meshes to approximate the Caputo time-fractional derivative, and use the LDG method to discretize the spatial derivative. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, several numerical experiments are provided which support the theoretical analysis. [ABSTRACT FROM AUTHOR]

The authors' concern of the present paper is to conduct a systematic study on a time-fractional nonlinear water wave equation which is an evolutionary version of the Boussinesq system. The study goes on by adopting a new analytical method based on the Laplace transform and the homotopy analysis method to the governing model and obtaining its approximate solutions in the presence of the Caputo fractional derivative. To analyze the influence of the Caputo operator on the dynamical behavior of the approximate solutions, some graphical illustrations in two- and three-dimensions are formally presented. Furthermore, several numerical tables are given to support the performance of the new analytical method in handling the time-fractional nonlinear water wave equation. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the numerical computation for a class of fourth-order fractional sub-diffusion equations with spatially variable coefficient under the first Dirichlet boundary conditions. Two finite difference schemes with second order accuracy are derived by applying L 2 − 1 σ formula and FL 2 − 1 σ formula respectively to approximate the time Caputo derivative. The main novelty is that a novel technique is introduced to deal with the first Dirichlet boundary conditions, which is compatible with the main equation with spatially variable coefficient. The solvability, unconditional stability and convergence of both schemes are proved by using the discrete energy method and mathematical induction. A difference scheme for such problem with two dimensions is also proposed and analyzed. Numerical results show that the suggested schemes have the almost same accuracy and the FL 2 − 1 σ scheme can reduce the storage and computational cost significantly. • The variable coefficient time fractional fourth-order sub-diffusion equation is considered. • A novel technique is introduced to deal with the first Dirichlet boundary conditions. • The resulting schemes proved to be unconditionally stable and convergent. • The fast scheme can reduce the computational complexity significantly. • The unconditionally stability and convergence of suggested schemes are proved. [ABSTRACT FROM AUTHOR]