Fractional Partial Derivatives Research Articles

A Numerical Algorithm for the Caputo Tempered Fractional Advection-Diffusion Equation

By transforming the Caputo tempered fractional advection-diffusion equation into the Riemann–Liouville tempered fractional advection-diffusion equation, and then using the fractional-compact Grunwald–Letnikov tempered difference operator to approximate the Riemann–Liouville tempered fractional partial derivative, the fractional central difference operator to discritize the space Riesz fractional partial derivative, and the classical central difference formula to discretize the advection term, a numerical algorithm is constructed for solving the Caputo tempered fractional advection-diffusion equation. The stability and the convergence analysis of the numerical method are given. Numerical experiments show that the numerical method is effective.

Solitary wave solutions and traveling wave solutions for systems of time-fractional nonlinear wave equations via an analytical approach

This paper introduces a new approximate-analytical approach for solving systems of Fractional Nonlinear Partial Differential Equations (FNPDEs). However, the main advantage of this new approximate-analytical approach is to obtain the analytical solution for general systems of FNPDEs in forms of convergent series with easily computable components using Caputo fractional partial derivative. Moreover, the convergence theorem and error analysis of the proposed method are also shown. Solitary wave solutions and traveling wave solutions for the system of fractional dispersive wave equations and the system of fractional long water wave equations are successfully obtained. The numerical solutions are also obtained in forms of tables and graphs to confirm the accuracy and efficiency of the suggested method.

A fast numerical method for fractional partial differential equations

In this paper, we use operational matrices of Chebyshev polynomials to solve fractional partial differential equations (FPDEs). We approximate the second partial derivative of the solution of linear FPDEs by operational matrices of shifted Chebyshev polynomials. We apply the operational matrix of integration and fractional integration to obtain approximations of (fractional) partial derivatives of the solution and the approximation of the solution. Then we substitute the operational matrix approximations in the FPDEs to obtain a system of linear algebraic equations. Finally, solving this system, we obtain the approximate solution. Numerical experiments show an exponential rate of convergence and hence the efficiency and effectiveness of the method.

On the Stability of a Hyperbolic Fractional Partial Differential Equation

In this paper, the $$\psi $$ -Riemann–Liouville fractional partial integral and the $$\psi $$ -Hilfer fractional partial derivative are introduced and some of its particular cases are recovered. Using the Gronwall inequality and these results, we investigate the Ulam–Hyers and Ulam–Hyers–Rassias stabilities of the solutions of a fractional partial differential equation of hyperbolic type in a Banach space $$({\mathbb {B}}, \left| \cdot \right| )$$ , real or complex. Finally, we present an example in order to elucidate the results obtained.

Fractional order solutions to fractional order partial differential equations

In this paper, we formulate a class of fractional order partial differential equations by concentrating on fractional order partial derivatives. We consider an initiative towards the formulation of fractional order Schrodinger equations, heat equation, and wave equation in the realm of fractional calculus. For a given fractional number, we obtain their respective fractional order solutions. In the light of fractional order quantum mechanics, we extend our analysis further for various potential functions for the case of the time-independent and time-dependent Schrodinger equations. Hereby, our proposal gives a refined solution to discrete dynamical systems. Namely, instead of taking the integral-valued steps in the evolution of a chosen dynamical system, it takes fractional-valued steps while performing an integral or solving a system of differential equations. Finally, concerning the extended analysis of our fractional order solutions, we anticipate interesting applications of the above formulation in other domains of physical sciences and natural systems.

New Wave Solutions of Time-Fractional Coupled Boussinesq–Whitham–Broer–Kaup Equation as A Model of Water Waves

The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.

Analytical Solutions for Nonlinear Systems of Conformable Space-Time Fractional Partial Differential Equations via Generalized Fractional Differential Transform

The present paper introduces approximate analytical solutions for nonlinear systems of conformable space-time fractional partial differential equations (CSTFPDEs) by using a generalized conformable fractional partial differential transform (GCFPDT). The GCFPDT is a modified version of the conformable fractional partial differential transform introduced in our recent work and it can be used as an efficient alternative transform to find analytical solutions for nonlinear systems of CSTFPDEs available in the literature. The convergence and error estimation of the proposed GCFPDT are also considered. Moreover, approximate analytical solutions to nonlinear system of gas dynamic equations, nonlinear system of KdV equations, and nonlinear system of approximate long water wave equations in the sense of conformable space-time fractional partial derivatives are successfully obtained to confirm the effectiveness and efficiency of the proposed GCFPDT.

The implicit midpoint method for Riesz tempered fractional diffusion equation with a nonlinear source term

In this paper, the implicit midpoint method is presented for solving Riesz tempered fractional diffusion equation with a nonlinear source term, where the tempered fractional partial derivatives are evaluated by the modified second-order Lubich tempered difference operator. Stability and convergence analyses of the numerical method are given. The numerical experiments demonstrate that the proposed method is effective.

New operational matrices of orthogonal Legendre polynomials and their operational

Many conventional physical and engineering phenomena have been identified to be well expressed by making use of the fractional order partial differential equations (FOPDEs). For that reason, a proficient and stable numerical method is needed to find the approximate solution of FOPDEs. This article is designed to develop the numerical scheme able to find the approximate solution of generalized fractional order coupled systems (FOCSs) with mixed partial derivative terms of fractional order. Our main objective in this article is the development of a new operational matrix for fractional mixed partial derivatives based on the orthogonal shifted Legendre polynomials (SLPs). The fractional derivatives are considered herein in the sense of Caputo. The proposed method has the advantage to reduce the considered problems to a system of algebraic equations which are simple in handling by any computational software. Being easily solvable, the associated algebraic system leads to finding the solution of the problem. Some examples are included to demonstrate the accuracy and validity of the proposed method.

Travelling wave solutions for fractional Korteweg-de Vries equations via an approximate-analytical method

This paper introduces an approximate-analytical method (AAM) for solving nonlinear fractional partial differential equations (NFPDEs) in full general forms. The main advantage of the paper is to apply the proposed AAM to solve the fractional Korteweg-de Vries (KdV) equations. Moreover, the analytical travelling wave solutions for the fractional KdV equation and the modified fractional KdV equation are successfully obtained. The numerical solutions are also obtained in the forms of tables and graphs. The fractional partial derivatives are considered in Caputo sense.

Edge-Based Fractional-Order Adaptive Strategies for Synchronization of Fractional-Order Coupled Networks With Reaction-Diffusion Terms.

In this paper, spatial diffusions are introduced to fractional-order coupled networks and the problem of synchronization is investigated for fractional-order coupled neural networks with reaction-diffusion terms. First, a new fractional-order inequality is established based on the Caputo partial fractional derivative. To realize asymptotical synchronization, two types of adaptive coupling weights are considered, namely: 1) coupling weights only related to time and 2) coupling weights dependent on both time and space. For each type of coupling weights, based on local information of the node's dynamics, an edge-based fractional-order adaptive law and an edge-based fractional-order pinning adaptive scheme are proposed. Furthermore, some new analytical tools, including the method of contradiction, L'Hopital rule, and Barbalat lemma are developed to establish adaptive synchronization criteria of the addressed networks. Finally, an example with numerical simulations is provided to illustrate the validity and effectiveness of the theoretical results.

A numerically efficient and conservative model for a Riesz space-fractional Klein–Gordon–Zakharov system

Departing from a fractional extension of the well-known one-dimensional Klein–Gordon–Zakharov system, we propose a numerically efficient model to approximate its solutions. The continuous model under investigation considers fractional derivatives of the Riesz type in space, with orders of differentiation in (1, 2]. In analogy with the non-fractional regime, the existence of a positive conserved energy quantity is established in this work. Motivated by this fact, the design of the numerical model focuses on the preservation of the energy. Using fractional-order centered differences to approximate the fractional partial derivatives, we propose a numerical model that preserves a positive discrete form of the energy. The existence and uniqueness of solutions are thoroughly established using fixed-point arguments, and the usual argument with Taylor polynomials is employed to prove the consistency of the numerical model. Some suitable bounds in terms of the energy invariants are found for the solutions of the numerical model. Moreover, using an extension of the energy method for fractional systems, we establish the stability and the convergence properties of the methodology. Finally, we provide some examples to illustrate the accuracy of our numerical implementation, including a numerical study of the convergence rate of the scheme that confirms the validity of the analytical results derived in this work.

Variational Problems with Partial Fractional Derivative: Optimal Conditions and Noether’s Theorem

In this paper, the necessary and sufficient conditions of optimality for variational problems with Caputo partial fractional derivative are established. Fractional Euler-Lagrange equations are obtained. The Legendre condition and Noether’s theorem are also presented.

Numerical computations of fractional nonlinear Hartmann flow with revised heat flux model

In this communication we have examined the magneto-hydrodynamic flow of viscoelastic liquid with revise version of thermal flux. Thermal conductivity of the fluid is variable. Effects of body forces such as gravity are also examined in terms of convection. Non-integer time derivatives are used for the better understanding of viscoelastic flow characteristics. In literature, these derivatives have been proved suitable for the predictions of hereditary systems such as viscoelastic flows. Heat transfer is analyzed in the flow field via modified fractional thermal gradient. Temperature gradient exists between the flow boundaries and increases with the passage of time. Flow is generated by the variable movement of flow boundary. All the segments of numerical modeling are completely addressed in this paper i.e. physical system, mathematical model, simulation, prediction, validation and verification. Unsteady motion of in-compressible fluid is directed by differential equations with fractional partial derivatives, for the solutions of these equations numerical techniques are necessary to use. Governing equations with appropriate flow conditions have been discretized by finite difference along with finite element algorithms. Theoretical error appraisals are hypothesized and created numerical plan has been approved by doing error investigations numerically regarding spatial directions. We have shown the estimate error and plotted the error curves in logarithmic scales of L2(Ω) and H1(Ω) norms by taking absolute approximations of the space discretization. Local Nusselt number and coefficients of skin friction are determined for non-integer viscoelastic model. Flow field is simulated for various values of physical parameters. The obtained results showed that fractional exponent α has opposite effects on the velocity, temperature profiles for time t=2. It is also noted that temperature gradient increases by the increase of fractional number α and thermal relaxation parameter γ. Viscoelastic flows in polymer and plastics industries can be addressed by similar technique.

On a Problem for a Mixed-Type Equation With Fractional Derivative

For a mixed-type equation with the Riemann–Liouville partial fractional derivative we study a problem where the boundary condition contains a linear combination of generalized fractional operators with the Gauss hypergeometric function. We find a solution to the considered problem explicitly by solving an equation with fractional derivatives of various orders and prove the uniqueness of the solution for various values of parameters of the mentioned operators.

Analytical solutions for conformable space-time fractional partial differential equations via fractional differential transform

This paper introduces an efficient fractional differential transform that is called “conformable fractional partial differential transform (CFPDT)” and its properties for solving linear and nonlinear conformable space-time fractional partial differential equations (CSTFPDEs). Moreover, a CFPDT is more practical and helpful for solving abroad CSTFPDEs. Analytical solutions to linear Navier–Stokes equation and nonlinear gas dynamic equations in sense of conformable space-time fractional partial derivatives are successfully obtained to confirm the accuracy and efficiency of the proposed transform.

A Fast Preconditioned Penalty Method for American Options Pricing Under Regime-Switching Tempered Fractional Diffusion Models

A fast preconditioned penalty method is developed for a system of parabolic linear complementarity problems (LCPs) involving tempered fractional order partial derivatives governing the price of American options whose underlying asset follows a geometry Levy process with multi-state regime switching. By means of the penalty method, the system of LCPs is approximated with a penalty term by a system of nonlinear tempered fractional partial differential equations (TFPDEs) coupled by a finite-state Markov chain. The system of nonlinear TFPDEs is discretized with the shifted Grunwald approximation by an upwind finite difference scheme which is shown to be unconditionally stable. Semi-smooth Newton’s method is utilized to solve the finite difference scheme as an outer iterative method in which the Jacobi matrix is found to possess Toeplitz-plus-diagonal structure. Consequently, the resulting linear system can be fast solved by the Krylov subspace method as an inner iterative method via fast Fourier transform (FFT). Furthermore, a novel preconditioner is proposed to speed up the convergence rate of the inner Krylov subspace iteration with theoretical analysis. With the above-mentioned preconditioning technique via FFT, under some mild conditions, the operation cost in each Newton’s step can be expected to be $$\mathcal{O}(N\mathrm{log}N)$$ , where N is the size of the coefficient matrix. Numerical examples are given to demonstrate the accuracy and efficiency of our proposed fast preconditioned penalty method.

New Analytical Technique for Solving a System of Nonlinear Fractional Partial Differential Equations

This paper introduces a new analytical technique (NAT) for solving a system of nonlinear fractional partial differential equations (NFPDEs) in full general set. Moreover, the convergence and error analysis of the proposed technique is shown. The approximate solutions for a system of NFPDEs are easily obtained by means of Caputo fractional partial derivatives based on the properties of fractional calculus. However, analytical and numerical traveling wave solutions for some systems of nonlinear wave equations are successfully obtained to confirm the accuracy and efficiency of the proposed technique. Several numerical results are presented in the format of tables and graphs to make a comparison with results previously obtained by other well-known methods.

A fast preconditioned policy iteration method for solving the tempered fractional HJB equation governing American options valuation

A fast preconditioned policy iteration method is proposed for the Hamilton–Jacobi–Bellman (HJB) equation involving tempered fractional order partial derivatives, governing the valuation of American options whose underlying asset follows exponential Lévy processes. An unconditionally stable upwind finite difference scheme with shifted Grünwald approximation is first developed to discretize the established HJB equation under the tempered fractional diffusion models. Next, the policy iteration method as an outer iterative method is utilized to solve the discretized HJB equation and proven to be convergent in finite steps to its numerical solution. Given the Toeplitz-like structure of the coefficient matrix in each policy iteration, the resulting linear system can be fast solved by the Krylov subspace method as an inner iterative method via fast Fourier transform (FFT). Furthermore, a novel preconditioner is proposed to speed up the convergence rate of the inner Krylov subspace iteration with theoretical analysis to ensure the linear system can be solved in O(NlogN) operations under some mild conditions, where N is the number of spatial node points. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed fast preconditioned policy method.

Partial, Composite Fractional Operators, and Their Properties and Applications

The paper discusses the properties of the partial fractional integrals, the partial fractional derivatives, and the composite fractional integrals and derivatives. Some basic formulas are derived; and the relations between the Riemann-Liouville and Caputo ones are also presented. Finally, as the specific application, a two dimensional Abel integral equation is solved.