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The fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Laplace homotopy perturbation method is discussed to obtain the approximate analytical solution of space-fractional and time-fractional Burgers equations. The method used combines the Laplace transform and the homotopy perturbation method. Numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional orders.

In this paper, we explore the existence and uniqueness theorem for a problem of the fractional Cauchy form, with dependence on the generalized proportional Caputo derivative. Furthermore, a new numerical technique is presented based on a decomposition formula for the generalized proportional Caputo derivative. Convergence analysis of the proposed technique is proved. Finally, numerical results are obtained to confirm the validity of the proposed method. [ABSTRACT FROM AUTHOR]

Our motive in this contribution is to find out the operational matrix of fractional derivative having non singular kernel namely Rabotnov fractional-exponential (RFE) kernel which is recently introduced and seeking numerical solution of non-linear Lienard equation which have Rabotnov fractional-exponential kernel fractional derivative. First we derive an approximation formula of the fractional order derivative of polynomial function z k in term of RFE kernel. Using this formula and some properties of shifted Legendre polynomials, we find out the operational matrix of fractional order differentiation. In the author of knowledge this operational matrix of RFE kernel fractional derivative is derived first time. We solve a new class of fractional partial differential equation (FPDEs) by implementation of this newly derived operational matrix. We show that our newly derived operational matrix is valid by taking an fractional derivative of a polynomial. Also, we study a new model of Lienard equation with RFE kernel fractional derivative and we can easily predict the feasibility of our numerical method to this new model. [ABSTRACT FROM AUTHOR]

In this paper, we study a coupled system of generalized Sturm–Liouville problems and Langevin fractional differential equations described by Atangana–Baleanu–Caputo (ABC for short) derivatives whose formulations are based on the notable Mittag-Leffler kernel. Prior to the main results, the equivalence of the coupled system to a nonlinear system of integral equations is proved. Once that has been done, we show in detail the existence–uniqueness and Ulam stability by the aid of fixed point theorems. Further, the continuous dependence of the solutions is extensively discussed. Some examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we present a new fractional-order mathematical model for a tumor-immune surveillance mechanism. We analyze the interactions between various tumor cell populations and immune system via a system of fractional differential equations (FDEs). An efficient numerical procedure is suggested to solve these FDEs by considering singular and nonsingular derivative operators. An optimal control strategy for investigating the effect of chemotherapy treatment on the proposed fractional model is also provided. Simulation results show that the new presented model based on the fractional operator with Mittag–Leffler kernel represents various asymptomatic behaviors that tracks the real data more accurately than the other fractional- and integer-order models. Numerical simulations also verify the efficiency of the proposed optimal control strategy and show that the growth of the naive tumor cell population is successfully declined. [ABSTRACT FROM AUTHOR]

Realizing the behavior of the solution in the asymptotic situations is essential for repetitive applications in the control theory and modeling of the real-world systems. This study discusses a robust and definitive attitude to find the interval approximate asymptotic solutions of fractional differential equations (FDEs) with the Atangana-Baleanu (A-B) derivative. In fact, such critical tasks require to observe precisely the behavior of the noninterval case at first. In this regard, we initially shed light on the noninterval cases and analyze the behavior of the approximate asymptotic solutions, and then, we introduce the A-B derivative for FDEs under interval arithmetic and develop a new and reliable approximation approach for fractional interval differential equations with the interval A-B derivative to get the interval approximate asymptotic solutions. We exploit Laplace transforms to get the asymptotic approximate solution based on the interval asymptotic A-B fractional derivatives under interval arithmetic. The techniques developed here provide essential tools for finding interval approximation asymptotic solutions under interval fractional derivatives with nonsingular Mittag-Leffler kernels. Two cases arising in the real-world systems are modeled under interval notion and given to interpret the behavior of the interval approximate asymptotic solutions under different conditions as well as to validate this new approach. This study highlights the importance of the asymptotic solutions for FDEs regardless of interval or noninterval parameters. 1 --> a --> [ABSTRACT FROM AUTHOR]

In this paper, a numerical scheme based on hybrid Chelyshkov functions (HCFs) is presented to solve a class of fractional optimal control problems (FOCPs). To this end, by using the orthogonal Chelyshkov polynomials, the HCFs are constructed and a general formulation for their operational matrix of the fractional integration, in the Riemann–Liouville sense, is derived. This operational matrix together with HCFs are used to reduce the FOCP to a system of algebraic equations, which can be solved by any standard iterative algorithm. Moreover, the application of presented method to the problems with a nonanalytic dynamic system is investigated. Numerical results confirm that the proposed HCFs method can achieve spectral accuracy to approximate the solution of FOCPs. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to find the exact and approximate analytical solution of Nizhnik-Novikov-Veselov system which may be considered as a model for an incompressible fluid with newly defined conformable derivative by using $$G'/G$$ expansion method and homotopy analysis method (HAM) respectively. Authors used conformable derivative because of its applicability and lucidity. It is known that, the NNV system of equations is an isotropic Lax integrable extension of the well-known KdV equation and has physical significance. Also, NNV system of equations can be derived from the inner parameter-dependent symmetry constraint of the KP equation. Then the exact solutions obtained by using $$G'/G$$ expansion method are compared with the approximate analytical solutions attained by employing HAM. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the $(k, s)$ -fractional integral and differential operators involving k-Mittag-Leffler function $E_{k,\rho,\beta}^{\delta}(z)$ as its kernel. Also, we establish various properties of these operators. Further, we consider a number of certain consequences of the main results. [ABSTRACT FROM AUTHOR]

The numerical solution of a fractional optimal control problem having a quadratic performance index is proposed and analyzed. The performance index of the fractional optimal control problem is considered as a function of both the state and the control variables. The dynamic constraint is expressed as a fractional differential equation that includes an integer derivative in addition to the fractional derivative. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Based on the shifted Legendre orthonormal polynomials, we employ the operational matrix of fractional derivatives, the Legendre–Gauss quadrature formula and the Lagrange multiplier method for reducing such a problem into a problem consisting of solving a system of algebraic equations. The convergence of the proposed method is analyzed. For confirming the validity and accuracy of the proposed numerical method, a numerical example is presented along with a comparison between our numerical results and those obtained using the Legendre spectral-collocation method. [ABSTRACT FROM AUTHOR]

In this paper we treat a fractional bosonic, scalar and vectorial, time equation namely Duffin-Kemmer-Petiau Equation. The fractional variational principle was used, the fractional Euler-Lagrange equations were presented. The wave functions were determined and expressed in terms of Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

This paper presents approximate analytical solutions for the fractional-order Brusselator system using the variational iteration method. The fractional derivatives are described in the Caputo sense. This method is based on the incorporation of the correction functional for the equation. Two examples are solved as illustrations, using symbolic computation. The numerical results show that the introduced approach is a promising tool for solving system of linear and nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, a novel coronavirus (2019-nCov) mathematical model with modified parameters is presented. This model consists of six nonlinear fractional order differential equations. Optimal control of the suggested model is the main objective of this work. Two control variables are presented in this model to minimize the population number of infected and asymptotically infected people. Necessary optimality conditions are derived. The Grünwald–Letnikov nonstandard weighted average finite difference method is constructed for simulating the proposed optimal control system. The stability of the proposed method is proved. In order to validate the theoretical results, numerical simulations and comparative studies are given. [ABSTRACT FROM AUTHOR]

We investigate some new class of hybrid type fractional differential equations and inclusions via some nonlocal three-point boundary value conditions. Also, we provide some examples to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we solve a system of fractional differential equations within a fractional derivative involving the Mittag-Leffler kernel by using the spectral methods. We apply the Chebyshev polynomials as a base and obtain the necessary operational matrix of fractional integral using the Clenshaw-Curtis formula. By applying the operational matrix, we obtain a system of linear algebraic equations. The approximate solution is computed by solving this system. The regularity of the solution investigated and a convergence analysis is provided. Numerical examples are provided to show the effectiveness and efficiency of the method. [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of fractional neutral impulsive differential equations with infinite delay via contraction mapping principle and fixed point technique for condensing map. We use the resolvent operator technique for integral equations to make the mild solution of the problem more appropriate. [ABSTRACT FROM AUTHOR]

The purpose of the current investigation is to generalize the concept of fractional derivative in the sense of Caputo–Fabrizio derivative (CF-derivative) for interval-valued function under uncertainty. The reason to choose this new approach is originated from the non-singularity property of the kernel that is critical to interpret the memory aftermath of the system, which was not precisely illustrated in the previous definitions. We study the properties of CF-derivative for interval-valued functions under generalized Hukuhara-differentiability. Then, the fractional differential equations under this notion are presented in details. We also study three real-world systems such as the falling body problem, Basset and Decay problem under interval-valued CF-differentiability. Our cases involve a demonstration that this new notion is accurately applicable for the mechanical and viscoelastic models based on the interval CF-derivative equations. [ABSTRACT FROM AUTHOR]

The current study aims to approximate the solution of fractional differential equations (FDEs) by using the fundamental properties of artificial neural networks (ANNs) for function approximation. In the first step, we derive an approximate solution of fractional differential equation (FDE) by using ANNs. In the second step, an optimization approach is exploited to adjust the weights of ANNs such that the approximated solution satisfies the FDE. Different types of FDEs including linear and nonlinear terms are solved to illustrate the ability of the method. In addition, the present scheme is compared with the analytical solution and a number of existing numerical techniques to show the efficiency of ANNs with high accuracy, fast convergence and low use of memory for solving the FDEs. [ABSTRACT FROM AUTHOR]