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This paper is concerned with the numerical approximations for the variable coefficient fourth-order fractional sub-diffusion equations subject to the second Dirichlet boundary conditions. We construct two effective difference schemes with second order accuracy in time by applying the second order approximation to the time Caputo derivative and the sum-of-exponentials approximation. By combining the discrete energy method and the mathematical induction method, the proposed methods proved to be unconditional stable and convergent. In order to overcome the possible singularity of the solution near the initial stage, a difference scheme based on non-uniform mesh is also given. Some numerical experiments are carried out to support our theoretical results. The results indicate that the our two main schemes has the almost same accuracy and the fast scheme can reduce the storage and computational cost significantly. [ABSTRACT FROM AUTHOR]

In this paper, efficient methods are derived for seeking a numerical solution to the time-fractional convection–diffusion equation whose solution very likely exhibits a weak regularity at the starting time. Here, the time-fractional derivative in the Caputo sense with order in (0 , 1) is discretized by the L 2 - 1 σ methods with uniform and non-uniform meshes and the spatial derivative is approximated by the local discontinuous Galerkin (LDG) finite element methods. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, some numerical examples are performed to testify the effectiveness of the obtained algorithm. [ABSTRACT FROM AUTHOR]

ntroduction Fractional calculus is an extension of derivatives and integrals to noninteger orders and it has been widely used to model engineering and scientific problems. Studying fractional calculus is an old topic in mathematical analysis, which goes back to Leibniz (1695) and Euler (1730). Exact solutions for the majority of fractional differential equations cannot be found easily, thus analytical and numerical methods must be used. In fact, several numerical methods for solving fractional differential equations have been presented in the literature and they have their own advantages and limitations. In recent years, the numerical solution of fractional equations has become a popular topic in applied sciences control and engineering. The Riccati equations play a significant role in many fields of engineering and applied science such as the theory of random processes, diffusion problems, transmission-line phenomena and optimal control theory. The Riccati differential equation is named after the Italian Nobleman Count Jacopo Francesco Riccati (1676-1754). In this work, we implement exponential spline method for solving the time fractional nonlinear Riccati initial value problem with the Caputo fractional derivative. Many authors have made attempts to solve these equations using different numerical methods. Some of these methods, such as the Laplace-Adomian-Pade method, the modified homotopy perturbation method, the He's variational iteration method, the Chebyshev wavelet operational matrix method, the finite difference method and Padevariational iteration method, the shifted Jacobi polynomial integral operational matrix method, The sine-cosine wavelet operational matrix, Variational iteration algorithm and the residual power series method. Material and methods There are several definitions of a fractional derivative of order such as Riemann-Liouville, Grunwald-Letnikove, and Caputo. In this paper, the Caputo fractional derivative is used for the formulation of the problem. The exponential spline function is presented for solving Riccati differential equations. Applying the continuity of the exponential spline function and its derivative we get the consistency relations. The relations of exponential spline function are utilized to transform the differential equations to the system of algebraic equations. Also, the error analysis of the exponential spline function bases is given. We use exponential spline approximation to develop a family of new numerical methods to obtain smooth approximations to the solution of fractional riccati differential equation. The proposed methods can be used to solve not only the classical Riccati differential equations but also the fractional ones. The convergence analysis of the method has been discussed. Some examples are included to demonstrate the validity and applicability of the technique and to compare the computed results with the other methods. Results and discussion Our main aim is to solve the Riccati differential equations by using exponential spline function. We propose two main methods to find an approximate solution of the fractional Riccati equation. The main advantage of our algorithm is that it can be used directly without using an assumption or transformation formulae and the convergence analysis of the method has been discussed. By using the truncation error to eliminate the coefficients of various powers h we can obtain classes of the methods. The presented method is its greater accuracy and smooth approximation without using an off-step point. The present method enables us to approximate the solution at every point of the interval [a, b]. Conclusion In this paper, the exponential spline has been applied to approximate the solution of the fractional riccati differential equations and we have obtained some methods of order-two. To demonstrate the convergence and applicability of the presented methods, some numerical experiments are reported. Comparisons with other methods and the exact solution show that this technique is a good approximation for solving the fractional Riccati differential equations. Based on the error analysis and convergence of these methods, we find that the order of convergence is two and applies to all kinds of fractional equations. [ABSTRACT FROM AUTHOR]

The generalized time‐fractional Fisher's equation is a substantial model for illustrating the system's dynamics. Studying effective numerical methods for this equation has considerable scientific importance and application value. In that direction, this paper presents designing and analyzing a high‐order numerical scheme for the generalized time‐fractional Fisher's equation. The time‐fractional derivative is taken in the Caputo sense and approximated using Euler backward discretization. The quasilinearization technique is used to linearize the problem, and then a compact finite difference scheme is considered for discretizing the equation in space direction. Our numerical method is convergent of Ok2−α+h4$$ O\left({k}^{2-\alpha }+{h}^4\right) $$, where h$$ h $$ and k$$ k $$ are step sizes in spatial and temporal directions, respectively. Three problems are tested numerically by implementing the proposed technique, and the acquired results reveal that the proposed method is suitable for solving this problem. [ABSTRACT FROM AUTHOR]

Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately evident that the method is fast and memory-efficient. Moreover, the method's design is such that good convergence properties may be expected. In this paper, we commence a systematic investigation of these convergence properties. [ABSTRACT FROM AUTHOR]

In this paper, an accurate technique is used to find an approximate solution to the fractional-order Duffing-Van der Pol (DVP, for short) oscillators equation which is reproducing kernel Hilbert space (RKHS, for short) method. The numerical results show that the n-term approximation is a rapidly convergent series representation and they present also the high accuracy and effectiveness of this method. The efficiency of the proposed method has been proved by the theoretical predictions and confirmed by the numerical experiments. [ABSTRACT FROM AUTHOR]

A numerical scheme is developed to solve the time-fractional linear Kuramoto-Sivahinsky equation in this work. The time-fractional derivative (of order γ) is taken in the Caputo sense. The scheme comprises the backward Euler formula in the temporal direction and the quintic B-spline collocation approach in the spatial direction. Through rigorous analysis, the proposed method is shown to be unconditionally stable and convergent of order 2 − γ and two in the temporal and spatial directions, respectively. Two test problems are solved numerically to demonstrate the convergence and accuracy of the method. [ABSTRACT FROM AUTHOR]

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order 0 < α < 1 , and 1 < β < 2 . The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered. [ABSTRACT FROM AUTHOR]

The well-known central finite difference approximation was combined with the trapezoid quadrature method in this study to provide a numerical solution of the linear system of Volterra integro-fractional differential equations (LSVI-FDEs) of arbitrary orders, where the fractional derivative is described in the Caputo sense and the orders are between zero and one. The method works by first using the central finite difference approximation to approximate the Caputo derivative at any fixed point and then using the trapezoidal rule to obtain a finite difference expression for our fractional equation, while recalling the linear spline approximation for the first steps. This new, more efficient method involves converting sets of equations and conditions into matrix relationships, from which symmetry matrices can be created in some cases. We also present a new approach for error analysis of the discrete numerical scheme and the explicit numerical technique for LSVI-FDEs. The multi-level explicit finite difference approximation's stability and convergence were explored, and a MatLab application was created to explain the results. Finally, several numerical examples are offered to demonstrate the technique's application. [ABSTRACT FROM AUTHOR]

Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately evident that the method is fast and memory-efficient. Moreover, the method’s design is such that good convergence properties may be expected. In this paper, we commence a systematic investigation of these convergence properties.