Jacobi Poly-fractonomials Research Articles

Efficient Solutions for Stochastic Fractional Differential Equations with a Neutral Delay Using Jacobi Poly-Fractonomials

This paper introduces a novel numerical technique for solving fractional stochastic differential equations with neutral delays. The method employs a stepwise collocation scheme with Jacobi poly-fractonomials to consider unknown stochastic processes. For this purpose, the delay differential equations are transformed into augmented ones without delays. This transformation makes it possible to use a collocation scheme improved with Jacobi poly-fractonomials to solve the changed equations repeatedly. At each iteration, a system of nonlinear equations is generated. Next, the convergence properties of the proposed method are rigorously analyzed. Afterward, the practical utility of the proposed numerical technique is validated through a series of test examples. These examples illustrate the method’s capability to produce accurate and efficient solutions.

Numerical approximation of fractional variational problems with several dependent variables using Jacobi poly-fractonomials

We discuss a new numerical scheme using Jacobi poly-fractonomials for the fractional variational problem (FVP) with several dependent variables. The FVP is defined using the Caputo fractional derivative. Using Jacobi poly-fractonomials in the discussed method, the considered FVP is reduced to a system of algebraic equations. By solving this system of algebraic equations, an approximate solution of FVP is accomplished. We also proved the convergence of the presented scheme and the fractional variational error. At last, we perform some figurative examples to exhibit the legitimacy and pertinence of the current method.

A Petrov-Galerkin finite element method using polyfractonomials to solve stochastic fractional differential equations

In this paper, we are concerned with existence, uniqueness and numerical approximation of the solution process to an initial value problem for stochastic fractional differential equation of Riemann-Liouville type. We propose and analyze a Petrov-Galerkin finite element method based on fractional (non-polynomial) Jacobi polyfractonomials as basis and test functions. Error estimates in L2 norm are derived and numerical experiments are provided to validate the theoretical results. As an illustrative application, we generate sample paths of the Riemann-Liouville fractional Brownian motion which is of importance in many applications ranging from geophysics to traffic flow in telecommunication networks.

A novel spectral Galerkin/Petrov–Galerkin algorithm for the multi-dimensional space–time fractional advection–diffusion–reaction equations with nonsmooth solutions

The usual classical polynomials-based spectral Galerkin and Petrov–Galerkin methods enjoy high-order accuracy for problems with smooth solutions. However, their accuracy and fidelity can be deteriorated when the solutions exhibit weakly singular behaviors and this issue becomes much more severe for polynomial-based spectral methods. The eigenfunctions of the Sturm–Liouville problems of fractional order serve as basis functions for constructing efficient spectral approximations for fractional differential models with nonsmooth solutions. In this paper, the Petrov–Galerkin spectral method is adopted to deal with the initial singularity in the temporal direction in which the first kind Jacobi poly-fractonomials are utilized as temporal trial functions and the second kind Jacobi poly-fractonomials as temporal test functions. Along the spatial direction, the Galerkin spectral method is adopted for the first time to deal with the boundary singularity in the spatial direction in which weighted Jacobi functions are utilized as bases in multi-dimensions. Various numerical experiments are provided to demonstrate the performance of the proposed schemes.

A-posteriori error estimations based on postprocessing technique for two-sided fractional differential equations

The analysis of diffusion-reaction equations with general two-sided fractional derivative characterized by a parameter p∈[0,1] is investigated in this paper. First, we present a Petrov-Galerkin method, derive a proper weak formulation and show the well-posedness of its weak solution. Moreover, on the basis of the two-sided Jacobi polyfractonomials, a priori error analysis of Petrov-Galerkin method is derived. Further, a posteriori error analysis is established rigorously. More precisely, we develop a novel postprocessing technique to enhance the Petrov-Galerkin method by adding a small amount of computation, and analyze asymptotically exact a-posteriori error estimators. Finally, we demonstrate the theoretical results with numerical examples.

A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

Fully spectral‐Galerkin method for the one‐ and two‐dimensional fourth‐order time‐fractional partial integro‐differential equations with a weakly singular kernel

AbstractIn the current paper, a space–time spectral‐Galerkin method is presented for the one‐ and two‐dimensional (1D & 2D) fourth‐order time‐fractional partial integro‐differential equation (TFPIDE) with a weakly singular kernel. In temporal direction, a Petrov‐Galerkin approach is used for discretization. Indeed, eigenfunctions of the first and second kind fractional Sturm‐Liouville problem (FSLP), which called Jacobi polyfractonomials, are used as temporal basis for the trial and test spaces, respectively. Also, spatial discretization is based on the Galerkin approximation with a combination of Legendre polynomials as basis. Fully discrete scheme, give rises to obtaining approximate solution of desired problem via solving a system of linear algebraic equations. Spectral accuracy and efficiency of the considered method are numerically demonstrated by some test problems with smooth and non‐smooth exact solutions in one‐ and two‐dimensional cases.

A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations

Fractional calculus and fractional-order modeling provide effective tools for modeling and simulation of anomalous diffusion with power-law scalings. In complex multi-fractal anomalous transport phenomena, distributed-order partial differential equations appear as tractable mathematical models, where the underlying derivative orders are distributed over a range of values, hence taking into account a wide range of multi-physics from ultraslow-to-standard-to-superdiffusion/wave dynamics. We develop a unified, fast, and stable Petrov–Galerkin spectral method for such models by employing Jacobi poly-fractonomials and Legendre polynomials as temporal and spatial basis/test functions, respectively. By defining the proper underlying distributed Sobolev spaces and their equivalent norms, we rigorously prove the well-posedness of the weak formulation, and thereby, we carry out the corresponding stability and error analysis. We finally provide several numerical simulations to study the performance and convergence of proposed scheme.

A Petrov-Galerkin spectral method for fractional convection–diffusion equations with two-sided fractional derivative

In this paper, we study a boundary value problem of fractional convection–diffusion equation with general two-sided fractional derivative. The well-posedness of the variation formulation is investigated under some properly assumptions. A Petrov-Galerkin method is developed, which employs the Jacobi poly-fractonomials for the trial and test space. The new approach allows the derivation of optimal error estimates in properly weighted Sobolev space and the matrix of the leading term is diagonal. We show that even for smooth data, we only obtain algebraic convergence due to the regularity of the solution. Some numerical examples are presented to demonstrate the validity of our theoretical results.

Spectral Galerkin approximation of optimal control problem governed by Riesz fractional differential equation

In this paper we investigate spectral Galerkin approximation of optimal control problem governed by Riesz fractional differential equation with control integral constraint. First order optimality condition is derived and the regularity of control problem is discussed. Based on first discretize then optimize approach a spectral Galerkin approximation of the control problem is developed, where two-sided Jacobi polyfractonomials are used to approximate the state variable and variational discretization is used to discretize the control variable. A priori error analysis for state variable, adjoint state variable and control variable is presented. Numerical experiments are carried out to illustrate the theoretical findings.

A unified spectral method for FPDEs with two-sided derivatives; part I: A fast solver

We develop a unified Petrov–Galerkin spectral method for a class of fractional partial differential equations with two-sided derivatives and constant coefficients of the form Dt2τ0u+∑i=1d[cliaiDxi2μiu+crixiDbi2μiu]+γu=∑j=1d[κljajDxj2νju+κrjxjDbj2νju]+f, where 2τ∈(0,2), 2τ≠1, 2μi∈(0,1) and 2νj∈(1,2), in a (1+d)-dimensional space–time hypercube, d=1,2,3,⋯, subject to homogeneous Dirichlet initial/boundary conditions. We employ the eigenfunctions of the fractional Sturm–Liouville eigen-problems of the first kind in [1], called Jacobi poly-fractonomials, as temporal bases, and the eigen-functions of the boundary-value problem of the second kind as temporal test functions. Next, we construct our spatial basis/test functions using Legendre polynomials, yielding mass matrices being independent of the spatial fractional orders (μi,νj,i,j=1,2,⋯,d). Furthermore, we formulate a novel unified fast linear solver for the resulting high-dimensional linear system based on the solution of generalized eigen-problem of spatial mass matrices with respect to the corresponding stiffness matrices, hence, making the complexity of the problem optimal, i.e., O(Nd+2). We carry out several numerical test cases to examine the CPU time and convergence rate of the method. The corresponding stability and error analysis of the Petrov–Galerkin method are carried out in [2].

A Spectral Method (of Exponential Convergence) for Singular Solutions of the Diffusion Equation with General Two-Sided Fractional Derivative

An open problem in the numerical solution of fractional partial differential equations (FPDEs) is how to obtain high-order accuracy for singular solutions; even for smooth right-hand sides solutions of FPDEs are singular. Here, we consider the one-dimensional diffusion equation with general two-sided fractional derivative characterized by a parameter $p\in [0,1]$; for $p=1/2$ we recover the Riesz fractional derivative, while for $p = 1$, 0 we obtain the one-sided fractional derivative. We employ a Petrov--Galerkin projection in a properly weighted Sobolev space with (two-sided) Jacobi polyfracnomials as basis and test functions. In particular, we derive these two-sided Jacobi polyfractonomials as eigenfunctions of a Sturm--Liouville problem with weights uniquely determined by the parameter $p$. We provide a rigorous analysis and obtain optimal error estimates that depend on the regularity of the forcing term, i.e., for smooth data (corresponding to singular solutions) we obtain exponential convergence, wh...

A Petrov–Galerkin spectral element method for fractional elliptic problems

We develop a new C0-continuous Petrov–Galerkin spectral element method for one-dimensional fractional elliptic problems of the form 0Dxαu(x)−λu(x)=f(x), α∈(1,2], subject to homogeneous boundary conditions. We employ the standard (modal) spectral element basis and the Jacobi poly-fractonomials as the test functions (Zayernouri and Karniadakis (2013)). We formulate a new procedure for assembling the global linear system from elemental (local) mass and stiffness matrices. The Petrov–Galerkin formulation requires performing elemental (local) construction of mass and stiffness matrices in the standard domain only once. Moreover, we efficiently obtain the non-local (history) stiffness matrices, in which the non-locality is presented analytically for uniform grids. We also investigate two distinct choices of basis/test functions: (i) local basis/test functions, and (ii) local basis with global test functions. We show that the former choice leads to a better-conditioned system and accuracy, while the latter results in ill-conditioned linear systems, and therefore, that is not an efficient and a proper choice of test function. We consider smooth and singular solutions, where the singularity can occur at boundary points as well as in the interior domain. We also construct two non-uniform grids over the whole computational domain in order to capture singular solutions. Finally, we perform a systematic numerical study of non-local effects via full and partial history fading in order to further enhance the efficiency of the scheme.

Petrov--Galerkin and Spectral Collocation Methods for Distributed Order Differential Equations

Distributed order fractional operators offer a rigorous tool for mathematical modelling of multiphysics phenomena, where the differential orders are distributed over a range of values rather than being just a fixed integer/fraction as in standard/fractional ODEs/PDEs. We develop two spectrally accurate schemes, namely, a Petrov--Galerkin spectral method and a spectral collocation method for distributed order fractional differential equations. These schemes are developed based on the fractional Sturm--Liouville eigen-problems (FSLPs) [M. Zayernouri and G. E. Karniadakis, J. Comput. Phys., 47 (2013), pp. 2108--2131]. In the Petrov--Galerkin method, we employ fractional (nonpolynomial) basis functions, called Jacobi polyfractonomials, which are the eigenfunctions of the FSLP of first kind, while we employ another space of test functions as the span of polyfractonomial eigenfunctions of the FSLP of second kind. We define the underlying distributed Sobolev space and the associated norms, where we carry out the...

Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions

In this paper, a nonpolynomial-based spectral collocation method and its well-conditioned variant are proposed and analyzed. First, we develop fractional differentiation matrices of nodal Jacobi polyfractonomials [M. Zayernouri and G. E. Karniadakis, J. Comput. Phys., 252 (2013), pp. 495--517] and generalized Jacobi functions [S. Chen, J. Shen, and L. L. Wang, Math. Comp., 85 (2016), pp. 1603--1638] on Jacobi--Gauss--Lobatto (JGL) points. We show that it suffices to compute the matrix of order $\mu\in (0,1)$ to compute that of any order $k +\mu$ with integer $k \geq 0$. With a different definition of the nodal basis, our approach also fixes a deficiency of the polyfractonomial fractional collocation method in [M. Zayernouri and G. E. Karniadakis, SIAM J. Sci. Comput., 38 (2014), pp. A40--A62]. Second, we provide explicit and compact formulas for computing the inverse of direct fractional differential collocation matrices at “interior” points by virtue of fractional JGL Birkhoff interpolation. This leads to optimal integration preconditioners for direct fractional collocation schemes and results in well-conditioned collocation systems. Finally, we present a detailed analysis of the singular behavior of solutions to rather general fractional differential equations (FDEs). Based upon the result, we have the privilege to adjust an index in our nonpolynomial approximation. Furthermore, by using the result, a rigorous convergence analysis is conducted by transforming an FDE into a Volterra (or mixed Volterra--Fredholm) integral equation.

Tempered Fractional Sturm--Liouville EigenProblems

Continuum-time random walk is a general model for particle kinetics, which allows for incorporating waiting times and/or non-Gaussian jump distributions with divergent second moments to account for Lévy flights. Exponentially tempering the probability distribution of the waiting times and the anomalously large displacements results in tempered-stable Lévy processes with finite moments, where the fluid (continuous) limit leads to the tempered fractional diffusion equation. The development of fast and accurate numerical schemes for such nonlocal problems requires a new spectral theory and suitable choice of basis functions. In this study, we introduce two classes of regular and singular tempered fractional Sturm--Liouville problems of two kinds (TFSLP-I and TFSLP-II) of order $\nu \in (0,2)$. In the regular case, the corresponding tempered differential operators are associated with tempering functions $p_I(x) = \exp(2\tau) $ and $p_{II}(x) = \exp(-2\tau)$, $\tau \geq 0$, respectively, in the regular TFSLP-I and TFSLP-II, which do not vanish in $[-1,1]$. In contrast, the corresponding differential operators in the singular setting are associated with different forms of $p_I(x) = \exp( 2\tau) (1-x)^{1+\alpha} (1+x)^{1+\beta}$ and $p_{II}(x) = \exp( -2\tau) (1-x)^{1+\alpha} (1+x)^{1+\beta}$, vanishing at $x=\pm 1$ in the singular TFSLP-I and TFSLP-II, respectively. The aforementioned tempered fractional differential operators are both of tempered Riemann--Liouville and tempered Caputo type of fractional order $\mu = \nu/2 \in (0,1)$. We prove the well-posedness of the boundary-value problems and that the eigenvalues of the regular tempered problems are real-valued and the corresponding eigenfunctions are orthogonal. Next, we obtain the explicit eigensolutions to TFSLP-I and -II as nonpolynomial functions, which we define as tempered Jacobi poly-fractonomials. These eigenfunctions are orthogonal with respect to the weight function associated with TFSLP-I and -II. Finally, we introduce these eigenfunctions as new basis (and test) functions for spectrally accurate approximation of functions and tempered-fractional differential operators. To this end, we further develop a Petrov--Galerkin spectral method for solving tempered fractional ODEs, followed by the corresponding stability and convergence analysis, which validates the achieved spectral convergence in our simulations.

A unified Petrov–Galerkin spectral method for fractional PDEs

Existing numerical methods for fractional PDEs suffer from low accuracy and inefficiency in dealing with three-dimensional problems or with long-time integrations. We develop a unified and spectrally accurate Petrov–Galerkin (PG) spectral method for a weak formulation of the general linear Fractional Partial Differential Equations (FPDEs) of the form 0Dt2τu+∑j=1dcj[ajDxj2μju]+γu=f, where 2τ, μj∈(0,1), in a (1+d)-dimensional space–time domain subject to Dirichlet initial and boundary conditions. We perform the stability analysis (in 1-D) and the corresponding convergence study of the scheme (in multi-D). The unified PG spectral method applies to the entire family of linear hyperbolic-, parabolic- and elliptic-like equations. We develop the PG method based on a new spectral theory for fractional Sturm–Liouville problems (FSLPs), recently introduced in Zayernouri and Karniadakis (2013). Specifically, we employ the eigenfunctions of the FSLP of first kind (FSLP-I), called Jacobi poly-fractonomials, as temporal/spatial bases. Next, we construct a different space for test functions from poly-fractonomial eigenfunctions of the FSLP of second kind (FSLP-II). Besides the high-order spatial accuracy of the PG method, we demonstrate its efficiency and spectral accuracy in time-integration schemes for solving time-dependent FPDEs as well, rather than employing algebraically accurate traditional methods, especially when 2τ=1. Finally, we formulate a general fast linear solver based on the eigenpairs of the corresponding temporal and spatial mass matrices with respect to the stiffness matrices, which reduces the computational cost drastically. We demonstrate that this framework can reduce to hyperbolic FPDEs such as time- and space-fractional advection (TSFA), parabolic FPDEs such as time- and space-fractional diffusion (TSFD) model, and elliptic FPDEs such as fractional Helmholtz/Poisson equations with the same ease and cost. Several numerical tests confirm the efficiency and spectral convergence of the unified PG spectral method for the aforementioned families of FPDEs. Moreover, we demonstrate the computational efficiency of the new approach in higher-dimensions e.g., (1+3), (1+5) and (1+9)-dimensional problems.

Spectral and Discontinuous Spectral Element Methods for Fractional Delay Equations

We first develop a spectrally accurate Petrov--Galerkin spectral method for fractional delay differential equations (FDDEs). This scheme is developed based on a new spectral theory for fractional Sturm--Liouville problems (FSLPs), which has been recently presented in [M. Zayernouri and G. E. Karniadakis, J. Comput. Phys., 252 (2013), pp. 495--517]. Specifically, we obtain solutions to FDDEs in terms of new nonpolynomial basis functions, called Jacobi polyfractonomials, which are the eigenfunctions of the FSLP of the first kind (FSLP-I). Correspondingly, we employ another space of test functions as the span of polyfractonomial eigenfunctions of the FSLP of the second kind (FSLP-II). We prove the wellposedness of the problem and carry out the corresponding stability and error analysis of the PG spectral method. In contrast to standard (nondelay) fractional differential equations, the delay character of FDDEs might induce solutions, which are either nonsmooth or piecewise smooth. In order to effectively trea...

Exponentially accurate spectral and spectral element methods for fractional ODEs

Current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy. Here, we present different methods for fractional ODEs that lead to exponentially fast decay of the error. First, we develop a Petrov–Galerkin (PG) spectral method for Fractional Initial-Value Problems (FIVPs) of the form Dtν0u(t)=f(t) and Fractional Final-Value Problems (FFVPs) DTνtu(t)=g(t), where ν∈(0,1), subject to Dirichlet initial/final conditions. These schemes are developed based on a new spectral theory for fractional Sturm–Liouville problems (FSLPs), which has been recently developed in [1]. Specifically, we obtain solutions to FIVPs and FFVPs in terms of the new fractional (non-polynomial) basis functions, called Jacobi polyfractonomials, which are the eigenfunctions of the FSLP of first kind (FSLP-I). Correspondingly, we employ another space of test functions as the span of polyfractonomial eigenfunctions of the FSLP of second kind (FSLP-II). Subsequently, we develop a Discontinuous Spectral Method (DSM) of Petrov–Galerkin sense for the aforementioned FIVPs and FFVPs, where the basis functions do not satisfy the initial/final conditions. Finally, we extend the DSM scheme to a Discontinuous Spectral Element Method (DSEM) for efficient longer time-integration and adaptive refinement. In these discontinuous schemes, we employ the asymptotic eigensolutions to FSLP-I & -II, which are of Jacobi polynomial forms, as basis and test functions. Our numerical tests confirm the exponential/algebraic convergence, respectively, in p- and h-refinements, for various test cases with integer- and fractional-order solutions.

Fractional Sturm–Liouville eigen-problems: Theory and numerical approximation

We first consider a regular fractional Sturm–Liouville problem of two kinds RFSLP-I and RFSLP-II of order ν∈(0,2). The corresponding fractional differential operators in these problems are both of Riemann–Liouville and Caputo type, of the same fractional order μ=ν/2∈(0,1). We obtain the analytical eigensolutions to RFSLP-I & -II as non-polynomial functions, which we define as Jacobi poly-fractonomials. These eigenfunctions are orthogonal with respect to the weight function associated with RFSLP-I & -II. Subsequently, we extend the fractional operators to a new family of singular fractional Sturm–Liouville problems of two kinds, SFSLP-I and SFSLP-II. We show that the primary regular boundary-value problems RFSLP-I & -II are indeed asymptotic cases for the singular counterparts SFSLP-I & -II. Furthermore, we prove that the eigenvalues of the singular problems are real-valued and the corresponding eigenfunctions are orthogonal. In addition, we obtain the eigen-solutions to SFSLP-I & -II analytically, also as non-polynomial functions, hence completing the whole family of the Jacobi poly-fractonomials. In numerical examples, we employ the new poly-fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results.