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This work proposes an iterative RBF-based method for solving nonlinear time-fractional Benjamin-Bona-Mahony-Burgers (BBMB) equations in arbitrary domains. The method relies on a newly developed approach for solving fractional ordinary differential equations (FODEs). To tackle the nonlinearity of the problem, we employ a simple quasi-linearization procedure. Subsequently, the solution is constructed using modified radial basis functions (RBFs), where the FODEs emerge automatically. The temporal domain is discretized using the Müntz polynomial basis, while the spatial domain discretization is achieved through the meshless backward substitution method. By conducting a series of problems in one, two, and three spatial dimensions, we verify the accuracy, efficiency, and robustness of the proposed scheme, and compare the computed results with references from the literature. The goal of this paper is to demonstrate the promising accuracy and efficiency of the proposed technique for solving time-fractional BBMB equations. [ABSTRACT FROM AUTHOR]

This paper proposes a method to solve a 2D time-fractional parabolic equation on arbitrary domains using an RBF partition of unity procedure based on a non-uniform time-stepping framework. The proposed method involves partitioning the main domain into overlapping subdomains, computing local approximations on each subdomain, and joining them together by PU weights to obtain a global approximation. This approach enhances stability, reduces computational costs, and replaces an ill-conditioned linear system with several small well-conditioned linear systems. To handle the weak singularity near the initial time, a time-graded mesh is used to obtain a non-uniform time-stepping scheme, which increases the temporal convergence rate. The convergence analysis of the method is examined to obtain an error bound for the local approximation. Numerical tests are conducted to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the splitting-up scheme for rough partial differential equations (RPDE) is introduced. The convergence rate of the splitting-up scheme for weak solutions to semilinear partial differential equations with linear rough noises is given. An application to stochastic partial differential equations with fractional Browian motions is considered. [ABSTRACT FROM AUTHOR]

This paper inspires us to propose an adaptive meshless numerical method of solving a 2D variable order time fractional mobile-immobile advection-diffusion equation on arbitrary domain. The mentioned method has three main contributions. (i) The adaptive meshless method could decrease amounts of computations effectively. (ii) On arbitrary domain, we successfully apply Legendre multiwaves in reproducing kernel Hilbert space (RKHS) defined on rectangle domain to approximate the exact solution. (iii) We skillfully build a special spline space, convergence order is obtained in view of convergence theories of spline space. Numerical experiments further demonstrate the validity of the method. • The adaptive meshless method could decrease amounts of computations effectively. • On arbitrary domain, we apply Legendre multiwaves to approximate the exact solution using some approximation theorems. • We successfully give the error estimate in the case of low regularity of the function. [ABSTRACT FROM AUTHOR]

In this paper, new and effective methods are provided for modeling and numerical simulation of viscoelastic plate, respectively. Viscoelastic plate is modeled by a variable fractional derivative model with better fitting effect and is numerically analyzed directly in time domain using shifted Chebyshev wavelets algorithm for the first time. A governing equation with three independent variables is established. The feasibility and accuracy of proposed algorithm are verified by convergence analysis and error estimations of mathematical example. Under different conditions, the ternary variable fractional differential equation of plate is solved numerically by shifted Chebyshev wavelets algorithm directly in time domain. The vibrations of plate are analyzed under different loads, loading times and temperatures. Numerical solutions for the stress of plate with two materials are also calculated and analyzed under the same load. The displacement of plate increases with increasing load, loading time and temperature. The bending resistance of PET plate is better than that of polyurea plate. These results are consistent with the existing references and actual practice. Therefore, the variable fractional model and shifted Chebyshev wavelets algorithm are accurate for the study of plate. They can be applied to complex problems in engineering. All obtained conclusions can provide theoretical basis for the protection and design of load-bearing structures in engineering. [ABSTRACT FROM AUTHOR]

An efficient strategy for the numerical solution of time-fractional diffusion-reaction problems is devised. A standard finite difference discretization of the space derivative is initially applied which results in a linear stiff term. Then a rectangular product-integration (PI) rule is implemented in an implicit-explicit (IMEX) framework in order to implicitly treat this linear stiff term and handle in an explicit way the non-linear, and usually non-stiff, term. The kernel compression scheme (KCS) is successively adopted to reduce the overload of computation and storage need for the persistent memory term. To reduce the computational effort the semidiscretized problem is described in a matrix-form, so as to require the solution of Sylvester equations only with small matrices. Theoretical results on the accuracy, together with strategies for the optimal selection of the main parameters of the whole method, are derived and verified by means of numerical experiments carried out in two-dimensional domains. The computational advantages with respect to other approaches are also shown and some applications to the detection of pattern formation are illustrated at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the control systems described by a large class of fractional semilinear parabolic equations. Firstly, we reformulate the fractional parabolic equations into abstract fractional differential equations associated with a semigroup on an appropriate Banach space. Secondly, we introduce a suitable concept on a mild solution for this kind of fractional parabolic equations and present the existence and uniqueness of mild solution by utilizing the theory of semigroup of linear operator, nonlinear analysis method and fixed point theorem. Then, the approximate controllability of the fractional semilinear parabolic equations is formulated and proved. At the end of the paper, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

The solution of d -dimensional fractional partial differential equations (FPDEs) is challenging in numerical computation. This paper proposes a low-rank preconditioned conjugate gradient (PCG) method for discretizations with a low-rank right-hand side stemming from the d -dimensional fractional Riesz equation. At each iteration step, a combination of low-rank in Tensor Train (TT) format technique and Fast Fourier Transform (FFT) is used to accelerate multilevel Toeplitz matrix-vector multiplications. Furthermore, a low-rank circulant preconditioner is constructed based on the spectrum of the coefficient matrix. The reported numerical results demonstrate the competitiveness of the new method compared with a state-of-the-art matrix-free approach based on FFT. [ABSTRACT FROM AUTHOR]

The goal of this paper is to introduce and study a new class of fractional differential hemivariational inequality formulated by an evolutionary hemivariational inequality and a fractional differential equation in Banach spaces. By employing the Rothe method and the surjectivity result, we derive the existence of unique solution for such a problem under some mild conditions. Moreover, we use the fully discrete scheme to approximate the fractional differential hemivariational inequality and provide an error estimate for the approximation. Finally, the main results are applied to obtain the unique solvability as well as the numerical analysis for a viscoelastic frictional contact problem with adhesion. [ABSTRACT FROM AUTHOR]

This paper is devoted to solving boundary value problems for important fractional differential equations of the Fokker–Planck family, in particular, to studying fractional differential equation for advection–dispersion. The consideration is carried out by the separation of variables (the Fourier method). Most part of this paper is devoted to justification of this method, to proof of a basis of the system of eigenfunctions for the basic equation (Aleroev et al., 2015) for modeling the random walk of a point particle which starts to move at the origin of coordinates in t = 0 on a self-similar fractal set. [ABSTRACT FROM AUTHOR]

In this paper, we develop an efficient spectral GJF-Petrov–Galerkin algorithm and the postprocessed method to solve a class of fractional initial value problems. The main part of these algorithms is to use a special set of general Jacobi functions (GJFs) to form the trial space and test space. We give a rigorous error analysis in non-uniformly weighted Sobolev spaces and obtain optimal error estimates. In particular, the postprocessing technique is used to construct the postprocessed method. In addition, we derive its superconvergence estimates and define a-posteriori error estimators that are asymptotically exact. Numerical experiments are included to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of wavelet approximation scheme based on Legendre and Chebyshev wavelets for finding the approximate solutions of distributed order linear differential equations. The operational matrix for distributed order fractional differential operator is derived for Legendre and Chebyshev wavelets basis. Furthermore, the obtained operational matrix along with Gauss Legendre quadrature formula and standard Tau method are utilized to reduce the distributed order linear differential equations into the system of linear algebraic equations. For the better understanding of the method, numerical algorithms are also provided for the considered problems. In order to verify the desired accuracy of the proposed method, five test examples are included and numerical experiments confirm the theoretical results and illustrate applicability and efficiency of the proposed method. Also, the convergence analysis, error bounds, error estimate and numerical stability of presented method via Legendre and Chebyshev wavelets are investigated. Moreover, comparison of the numerical results obtained by the proposed approach is provided with the results of existing method. [ABSTRACT FROM AUTHOR]

In this paper we focus on the subdiffusive Black–Scholes (B–S) model. The main part of our work consists of the finite difference method as a numerical approach to the option pricing in the considered model. We find the governing fractional differential equation and the related weighted numerical scheme being a generalization of the classical Crank–Nicolson (C–N) scheme. The proposed method has 2 − α order of accuracy with respect to time where α ∈ (0 , 1) is the subdiffusion parameter, and 2 with respect to space. Further, we provide the stability and convergence analysis. Finally, we present some numerical results. [ABSTRACT FROM AUTHOR]

In recent years, fractional partial differential equation (FPDE) has been widely applied in options pricing problems, which better explains many important empirical facts of financial markets. However, the vast majority of the literatures focus on pricing the single asset option under the FPDE framework. In this paper, a two-dimensional FPDE governing the valuation of rainbow options is established, where two underlying assets are assumed to follow independent exponential Lévy processes, and its boundary conditions are determined by solving one-dimensional FPDEs. A second-order accurate finite difference scheme is proposed to discretize the two-dimensional FPDE. Given the block Toeplitz with Toeplitz block structure of the coefficient matrix, a fast preconditioned Krylov subspace method is developed for solving the resulting linear system with O (N log N) computational complexity per iteration, where N is the matrix size. The proposed preconditioner accelerates the convergence of the iterative method with theoretical analysis. Numerical examples are given to demonstrate the accuracy and efficiency of our proposed numerical methods. [ABSTRACT FROM AUTHOR]

The quenching phenomenon and its physical characters are critical issues in the study of combustion process. In this paper, a two-dimensional temporal fractional combustion model with Caputo derivative is considered. We design an ingenious adaptive finite difference discontinuous Galerkin method to solve this fractional combustion model. To catch quenching moment more accurately, an explicit bisection adaptive strategy is proposed for discretization of fractional derivative. Moreover, discontinuous Galerkin method is employed for spatial discretization to settle large gradient changes and non-smoothness property of solution in adjacent area of quenching moment. The physical properties of combustion process, such as positivity, monotonicity and stability, are theoretically or numerically investigated. Finally, we carry out systematic numerical experiments for such abstract combustor with different shapes and various parameter settings, which illustrate that the dynamics of such model depends on the intensity of initial inputs, the order of fractional derivative and the area of spatial domain. [ABSTRACT FROM AUTHOR]

Abstract: There has been a recent increase in interest in the use of distributed order differential equations (particularly in the case where the derivatives are given in the Caputo sense) to model various phenomena. Recent papers have provided insights into the numerical approximation of the solution, and some results on existence and uniqueness have been proved. In each case, the representation of the solution depends, among other parameters, on Caputo-type initial conditions. In this paper we discuss the existence and uniqueness of solutions and we propose a numerical method for their approximation in the case where the initial conditions are not known and, instead, some Caputo-type conditions are given away from the origin. [Copyright &y& Elsevier]

Abstract In this paper, a local compact integrated radial basis function (CIRBF) method is proposed to solve the time fractional convection–diffusion–reaction equations. The proposed CIRBF scheme derives high accuracy and fast convergence rate with grid refinement. The spatial derivatives are discretized by combining with the integrated RBF approximation and compact approximation in a stencil. The nodal function values and the nodal second derivatives are used to establish the relation between the physical space and the RBF weight space. The time discretization is replaced by the second order shifted Grünwald scheme. Moreover, the unconditional stability and convergence of the proposed method are shown. Finally, some numerical examples on 2D and 3D domains are carried out to verify the accuracy and efficiency of the proposed algorithm for the fractional differential equations. [ABSTRACT FROM AUTHOR]

Abstract In this paper we apply the tools of harmonic analysis to study the Cauchy problem for time fractional Schrödinger equations. The existence and a sharp decay estimate for solutions of the given problem in two different spaces are addressed. Some fundamental properties of operators appearing in the solution of the problem are also discussed. [ABSTRACT FROM AUTHOR]

Abstract The present paper deals with global existence of weak solutions of a time-space fractional Landau–Lifshitz–Bloch equation involving the weak Caputo derivative and a fractional Laplacian. We use Faedo–Galerkin method with some commutator estimates in order to prove global existence of weak solutions for the model. The uniqueness is also discussed in a special one dimensional case. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the fractional Choquard equation (− Δ) s u + u = (| x | − μ ∗ F (u)) f (u) , in R N , where N ≥ 3 , 0 < s < 1 , 0 < μ < min { N , 4 s } , and f ∈ C (R , R) satisfies the general Berestycki–Lions conditions. Combining constrained variational method with deformation lemma, we obtain a ground state solution of Pohoz̆aev type for the above equation. The result improves some ones in Shen et al. (2016). [ABSTRACT FROM AUTHOR]

Abstract This paper introduces a high-order time stepping scheme, that is based on using Fourier spectral in space and a fourth-order diagonal Padé approximation to the matrix exponential function for solving multi-dimensional space-fractional reaction–diffusion equations. The resulting time stepping scheme is developed based on an exponential time differencing approach such that it alleviates solving a large non-linear system at each time step while maintaining the stability of the scheme. The non-locality of the fractional operator in some other numerical schemes for these equations leads to full and dense matrices. This scheme is able to overcome such computational inefficiency due to the full diagonal representation of the fractional operator. It also attains spectral convergence for multiple spatial dimensions. The stability of the scheme is discussed through the investigation of the amplification symbol and plotting its stability regions, which provides an indication of the stability of the method. The convergence analysis is performed empirically to show that the scheme is fourth-order accurate in time, as expected. Numerical experiments on reaction–diffusion systems with application to pattern formation are discussed to show the effect of the fractional order in space-fractional reaction–diffusion equations and to validate the effectiveness of the scheme. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the existence of solutions for the nonlinear fractional differential equation with the boundary value conditions where and are the standard Caputo derivative with . By using the contraction mapping principle and the Schauder fixed point theorem, some existence results are obtained. In addition, Lemma 2.6 in this paper is a valuable tool in seeking solvability of the fractional differential equations. [Copyright &y& Elsevier]

Abstract In this paper, the invariance properties of the time fractional (2 + 1) -dimensional Zakharov–Kuznetsov modified equal width (ZK-MEW) equation have been investigated using the Lie group analysis method. Lie point symmetries of the time fractional (2 + 1) -dimensional ZK-MEW equation have been derived by using the Lie group analysis method of fractional differential equations. Using the Lie symmetry analysis, the vector fields and the symmetry reduction of this equation are obtained. It is shown that the time fractional (2 + 1) -dimensional ZK-MEW equation can be transformed to an equation with Erdélyi–Kober fractional derivative. Finally using new conservation theorem with formal Lagrangian, the new conserved vectors are well constructed with a detailed derivation, which constitutes the conservation analysis for the time fractional (2 + 1) -dimensional ZK-MEW equation. [ABSTRACT FROM AUTHOR]

Nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in Science and Engineering. However, solving these type of models is computationally demanding. In this paper, we propose an exponential integrator method for space fractional models as an attractive and easy-to-code alternative for other existing second-order exponential integrator methods. This scheme is based on using a real distinct poles discretization for the underlying matrix exponentials. One of the major benefits of the proposed scheme is that the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The scheme is established to be second-order convergent; and proven to be robust for nonlinear space fractional reaction–diffusion problems involving non-smooth initial data. Our approach is exhibited by solving a system of two-dimensional problems which exhibits pattern formation and has applications in cell-division. Empirically, super-diffusion processes are displayed by investigating the effect of the fractional power of the underlying Laplacian operator on the pattern formation found in these models. Furthermore, the superiority of our method over competing second order ETD schemes, BDF2 scheme, and IMEX schemes is demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity ε 2 s M ( [ u ] s , A ε 2 ) ( − Δ ) A ε s u + V ( x ) u = | u | 2 s ∗ − 2 u + h ( x , | u | 2 ) u , x ∈ R N , u ( x ) → 0 , as | x | → ∞ , where ( − Δ ) A ε s is the fractional magnetic operator with 0 < s < 1 , 2 s ∗ = 2 N ∕ ( N − 2 s ) , M : R 0 + → R + is a continuous nondecreasing function, V : R N → R 0 + and A : R N → R N are the electric and magnetic potentials, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that ε < E ; and (ii) for any m ∗ ∈ N , has m ∗ pairs of solutions if ε < E m ∗ , where E and E m ∗ are sufficiently small positive numbers. Moreover, these solutions u ε → 0 as ε → 0 . [ABSTRACT FROM AUTHOR]

In this paper, the auxiliary equation method is applied to obtain analytical solutions of (2 + 1)-dimensional time-fractional Zoomeron equation and the time-fractional third order modified KdV equation in the sense of the conformable fractional derivative. Given equations are converted to the nonlinear ordinary differential equations of integer order; and then, the resulting equations are solved using a novel analytical method called the auxiliary equation method. As a result, some exact solutions for them are successfully established. The exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Schrödinger equation ( − Δ ) s u + V ( x ) u = f ( x , u ) , x ∈ R N , where ( − Δ ) s ( s ∈ ( 0 , 1 ) ) denotes the fractional Laplacian. Under some weaker conditions on the nonlinearity, we obtain the existence of solutions for the above problem in periodic case and asymptotically periodic case via variational methods, respectively. [ABSTRACT FROM AUTHOR]

This paper presents a deep analysis of a time-dependent Schrödinger equation with fractional time derivative. After the discretization of the spatial operator the equation is reformulated in terms of a special system of fractional differential equations; it occurs that the eigenvalues of the coefficient matrix lay on the boundary of the stability region of fractional differential equations. The main difficulties in solving this system are hence related to the simultaneous presence of persisting oscillations (possibly with high frequency as it is typical with Schrödinger equations) and a persisting memory (as a consequence of the fractional order); moreover, an accurate spatial discretization gives rise to systems of large to very large size, involving a noteworthy computational complexity. By means of a theoretical analysis the exact solution is split into two or three terms (depending on the order of the fractional derivative), thus to face the numerical computation by different and suitably selected methods: direct evaluation of matrix functions for the terms characterized by smooth behaviour but with persistent memory and a step-by-step strategy, in conjunction with matrix function, for the oscillating term. In both cases, Krylov subspace methods are employed for the computation of matrix functions and convergence results are presented. [ABSTRACT FROM AUTHOR]

In this paper, we propose a hybrid Laplace transform and finite difference method to price (finite-maturity) American options, which is applicable to a wide variety of asset price models including the constant elasticity of variance (CEV), hyper-exponential jump–diffusion (HEJD), Markov regime switching models, and the finite moment log stable (FMLS) models. We first apply Laplace transforms to free boundary partial differential equations (PDEs) or fractional partial differential equations (FPDEs) governing the American option prices with respect to time, and obtain second order ordinary differential equations (ODEs) or fractional differential equations (FDEs) with free boundary, which is named as the early exercise boundary in the American option pricing. Then, we develop an iterative algorithm based on finite difference methods to solve the ODEs or FDEs together with the unknown free boundary values in the Laplace space. Both the early exercise boundary and the prices of American options are recovered through inverse Laplace transforms. Numerical examples demonstrate the accuracy and efficiency of the method in CEV, HEJD, Markov regime switching models and the FMLS models. [ABSTRACT FROM AUTHOR]

In this paper, a multi-dimensional α -fractional diffusion–wave equation is introduced and the properties of its fundamental solution are studied. This equation can be deduced from the basic continuous time random walk equations and contains the Caputo time-fractional derivative of the order α / 2 and the Riesz space-fractional derivative of the order α so that the ratio of the derivatives orders is equal to one half as in the case of the conventional diffusion equation. It turns out that the α -fractional diffusion–wave equation inherits some properties of both the conventional diffusion equation and of the wave equation. In particular, in the one- and two-dimensional cases, the fundamental solution to the α -fractional diffusion–wave equation can be interpreted as a probability density function and the entropy production rate of the stochastic process governed by this equation is exactly the same as the case of the conventional diffusion equation. On the other hand, in the three-dimensional case this equation describes a kind of anomalous wave propagation with a time-dependent propagation phase velocity. [ABSTRACT FROM AUTHOR]

In this paper, we established some sufficient conditions for oscillation of solutions of a class of impulsive partial fractional differential equations with forcing term subject to Robin and Dirichlet boundary conditions by using differential inequality method. As an application, we included an example to illustrate the main result. [ABSTRACT FROM AUTHOR]

In this paper, a high-order local discontinuous Galerkin (LDG) method combined with weighted and shifted Grünwald difference (WSGD) approximation is developed and discussed for a Caputo time-fractional subdiffusion equation. The time fractional derivative of order α , 0 < α < 1 , is approximated by a third-order method based on the idea of WSGD operator, while the spatial operator is approximated by the LDG method. Some useful lemmas are first introduced and proved, then the analysis of stability and optimal error estimate O ( Δ t 3 + h k + 1 ) are obtained for the LDG method. Extensive numerical results using P k , k = 0 , 1 , 2 , 3 , elements are presented to validate our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, by the critical point theory, we consider the existence and multiplicity of solutions for the following fractional differential equation t D ∞ α ( − ∞ D t α u ( t ) ) + L ( t ) u ( t ) = ∇ W ( t , u ( t ) ) , t ∈ R , where α ∈ ( 1 2 , 1 ] , − ∞ D t α and t D ∞ α are left and right Liouville–Weyl fractional derivatives of order α on the whole axis R respectively, u ∈ R n , L ( t ) is positive definite symmetric matrix for all t ∈ R and W : R × R n → R is a suitably chosen function. [ABSTRACT FROM AUTHOR]

This paper focuses on providing the computation methods for the backward time tempered fractional Feynman–Kac equation, being one of the models recently proposed in Wu et al. (2016). The discretization for the tempered fractional substantial derivative is derived, and the corresponding finite difference and finite element schemes are designed with well established stability and convergence. The performed numerical experiments show the effectiveness of the presented schemes. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical solution of the time-fractional diffusion-wave equation. Essentially, the time fractional diffusion-wave equation differs from the standard diffusion-wave equation in the time derivative term. We propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of Jacobi interpolation polynomial for this equation. The convergence of the method is rigorously established. Numerical tests are carried out to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of global in time solutions of a time fractional reaction–diffusion system with time fractional derivatives. Furthermore, the large time behavior of bounded solutions is investigated. Our method of proof relies on a maximal regularity result for fractional linear reaction–diffusion equations that has been derived by Bajlekova (2001). [ABSTRACT FROM AUTHOR]

In this paper, we deal with the Navier–Stokes equations with the time-fractional derivative of order α ∈ ( 0 , 1 ) , which can be used to simulate anomalous diffusion in fractal media. We firstly give the concept of the weak solutions and establish the existence criterion of weak solutions by means of Galerkin approximations in the case that the dimension n ≤ 4 . Moreover, a complete proof of the uniqueness is given when n = 2 . At last we give a sufficient condition of optimal control pairs. [ABSTRACT FROM AUTHOR]

This paper deals with an initial–boundary value problem for a distributed order time-fractional diffusion equation with the fractional derivative in the Caputo sense. The method of the eigenfunctions expansion in combination with the Laplace transform is first employed to prove the uniqueness and existence of the solution to the initial–boundary value problem and then to show its analyticity in time. As an application of the analyticity of the solution, a uniqueness result for an important inverse problem of determination of the weight function in the distributed order derivative contained in the time-fractional diffusion equation from one interior point observation of its solution is obtained. [ABSTRACT FROM AUTHOR]

The traditional element free Galerkin (EFG) approach is constructed on variational weak form that the test and trial functions are shape functions of moving least squares (MLS) approximation. In the current paper, we propose a new version of the EFG method based on the shape functions of reproducing kernel particle method (RKPM). In other words, based on the developed approach in Han and Meng (2001) the fractional Tricomi-type equation will be solved using the new technique. The fractional derivative has been introduced in the Caputo’s sense and is approximated by a finite difference plan of order O ( τ 3 − α ) , 1 < α < 2 . We use the EFG-RKPM to discrete the spatial direction. We illustrate some numerical results on non-rectangular domains. The unconditional stability and convergence of the new technique have been proved. Numerical examples display the theoretical results and the efficiency of the proposed approach. Also, the numerical results are compared with the finite element method (FEM) and EFG-MLS procedure. [ABSTRACT FROM AUTHOR]

In this paper, a new method based on the Legendre wavelets expansion together with operational matrices of fractional integration and derivative of these basis functions is proposed to solve time-fractional seventh order KdV (sKdV) equation. Two-dimensional Legendre wavelet method is applied to compute the numerical solution of nonlinear time-fractional sKdV equation. The approximate solutions of nonlinear time fractional sKdV equation thus obtained by Legendre wavelet method are compared with the exact solutions as well as homotopy analysis method (HAM). The present scheme is very simple, effective and convenient for obtaining numerical solution of the sKdV equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse problem of determining a source in a time fractional diffusion equation where data are given at a fixed time. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. Using the method of truncated integration and the Fourier transform, we construct regularized solutions and derive explicitly error estimate. Two numerical examples are presented to illustrate the validity and effectiveness of our method. [ABSTRACT FROM AUTHOR]

As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena. In this paper, we develop an exponentially accurate Jacobi–Gauss–Lobatto collocation (J–GL-C) method to solve the variable-order fractional Schrödinger equations in one dimension (1D) and two dimensions (2D). In this method, the aforementioned problem is reduced to a system of ordinary differential equations (ODEs) in the time variable. As a result, we propose two efficient schemes for dealing with the numerical solutions of initial value problems for nonlinear system of ordinary differential equations, one based on the implicit Runge–Kutta (IRK) method of fourth order and the other based on Jacobi–Gauss–Radau collocation (J–GR-C) method. The validity and effectiveness of the two methods are demonstrated by solving three numerical examples in 1D and 2D. The convergence of the methods is graphically analyzed. The results demonstrate that the proposed methods are powerful algorithms with high accuracy for solving the variable-order nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a type of fractional diffusion equation (FDE) with variable coefficients on a finite domain. Firstly, we utilize a second-order scheme to approximate the Riemann–Liouville fractional derivative and present the finite difference scheme. Specifically, we discuss the Crank–Nicolson scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the scheme and conclude that the scheme is unconditionally stable and convergent with the second-order accuracy of O ( τ 2 + h 2 ) . Furthermore, we develop a fast accurate iterative method for the Crank–Nicolson scheme, which only requires storage of O ( m ) and computational cost of O ( m log m ) while retaining the same accuracy and approximation property as Gauss elimination, where m = 1 / h is the partition number in space direction. Finally, several numerical examples are given to show the effectiveness of the numerical method, and the results are in excellent agreement with the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper derives analytical solutions for a class of new multi-term fractional-order partial differential equations, which include the terms for spatial diffusion, time-fractional diffusion (multi-term) and reaction. These models can be used to describe the nonlinear relationship between the shear stress and shear rate of generalized viscoelastic Oldroyd-B fluid and Burgers fluid. By using a modified separation of variables method, the governing fractional-order partial differential equations are transformed into a set of fractional-order ordinary differential equations. Mikusiński-type operational calculus is then employed to obtain the exact solutions of the linear fractional ordinary differential equations with constant coefficients. The solutions are expressed in terms of multivariate Mittag-Leffler functions. Different situations for the unsteady flows of generalized Oldroyd-B fluid and Burgers fluid due to a moving plate are considered via examples. Integral representations of the solutions are presented. It is shown that the presented results reduce to the corresponding results for classical Navier–Stokes, Oldroyd-B, Maxwell and second-grade fluids as special cases. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following fractional Schrödinger equation with sublinear perturbation and steep potential well { ( − △ ) s u + λ V ( x ) u = f ( x , u ) + α ( x ) | u | ν − 2 u in R N , u ∈ H s ( R N ) , where 0 < s < 1 , 2 s < N , λ > 0 , 1 < ν < 2 , f ∈ C ( R N × R ) is of subcritical growth. By using variational methods, we prove that such a class of equations possess at least two nontrivial solutions. Moreover, the phenomenon of concentration of solutions is explored as well. [ABSTRACT FROM AUTHOR]

In this paper, the stability analysis is presented for a first order difference scheme applied to a nonhomogeneous time fractional Schrödinger differential equation. Based on the z-transform method, stability theorems are proved for the abstract case. The stability results are applied on initial–boundary value problems for multidimensional time fractional Schrödinger differential equations. Theoretical findings are validated by numerical experiments on one and two-dimensional time fractional Schrödinger differential equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the approximation of the solution for neutral fractional differential equation with nonlocal conditions in an arbitrary separable Hilbert space H . We study an associated integral equation and then, consider a sequence of approximate integral equations obtained by the projection of considered associated nonlocal neutral fractional integral equation onto finite dimensional space. The sufficient condition for the existence and uniqueness of solutions to every approximate integral equation is derived by using analytic semigroup and Banach fixed point theorem. We demonstrate convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Moreover, we consider the Faedo–Galerkin approximations of the solution and demonstrate some convergence results. An example is also provided to illustrate the discussed abstract theory. [ABSTRACT FROM AUTHOR]

In this paper, the multi-step differential transform method is employed for the first time to solve a time-fractional nonlinear Bloch system. The nonlinear Bloch equation is known to govern the dynamics of an ensemble of spins, controlling the basic process of nuclear magnetic resonance (NMR). This nonlinear Bloch equation is formed from a system of nonlinear ordinary differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The results obtained are in good agreement with the ones in the open literature and it is shown that the technique introduced here is robust, efficient and easy to implement. [ABSTRACT FROM AUTHOR]

In this paper, we present the Adomian decomposition method and its modifications combined with convergence acceleration techniques, such as the diagonal Padé approximants and the iterated Shanks transforms, to solve nonlinear fractional ordinary differential equations. Two nonlinear numeric examples demonstrate that either the diagonal Padé approximants or the iterated Shanks transforms can efficiently extend the effective convergence region of the decomposition series solution. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the first purpose is treating Cauchy problems and boundary value problems for nonlinear impulsive differential equations with Caputo fractional derivative. We introduce the concept of piecewise continuous solutions for impulsive Cauchy problems and impulsive boundary value problems respectively. By using a new fixed point theorem, we obtain many new existence, uniqueness and data dependence results of solutions via some generalized singular Gronwall inequalities. The second purpose is discussing Ulam stability for impulsive fractional differential equations. Some new concepts in stability of impulsive fractional differential equations are offered from different perspectives. Some applications of our results are also provided. [Copyright &y& Elsevier]