Product Integration Rules Research Articles

Numerical Simulations and FPGA Implementations of Fractional-Order Systems Based on Product Integration Rules

Product integration (PI) rules are well known numerical techniques that are used to solve differential equations of integer and, recently, fractional orders. Due to the high memory dependency of the PI rules used in solving fractional-order systems (FOS), their hardware implementation is very difficult and resources-demanding. In this paper, modified versions of the PI rules are introduced to facilitate their digital implementations. The studied rules are PI rectangular, PI trapezoidal, and predict-evaluate-correct-evaluate (PECE) rules. The three modified versions of the PI rules are validated using a benchmark system of differential equations for different sizes of the memory window to show the effect of the window size on the solution accuracy. Additionally, the three modified versions of the PI rules are used to simulate a novel fractional-order chaotic system (FOCS) where its bifurcation diagrams are discussed with the window length parameter. The chaotic system is then implemented on a Field Programmable Gate Array (FPGA). There are only a few trials in literature to implement the fractional PECE algorithm on FPGA, nevertheless, the proposed FPGA realization is compared with the most recent of these trials. The FPGA implementations of the three PI rules are made using Xilinx ISE 14.7 on Artix 7 kit. The achieved throughput are 1280 Mbits/sec for PI rectangular, 128.8 Mbits/sec for PI trapezoidal, and 129.12 Mbits/sec for fractional-order PECE.

Numerical Solutions of Conformable Fractional Differential Equations by Taylor and Finite Difference Methods

We drive efficient and reliable finite difference methods for fractional differential equations (FDEs) based on recently defined conformable fractional derivative. We first derive fractional Euler and fractional Taylor methods based on the fractional Taylor expansion. This fractional Taylor series are the generalized fractional Taylor series that are independent of initial point. We show that the proposed methods are more efficient and faster by applying these methods on first order FDEs and second order oscillatory FDEs. Our second approach is based on inverting FDEs to a weakly singular integral equation that is approximated by product integration rule. This new definition has no special functions and thus the proposed numerical methods will be more accurate and easier to implement than existing methods for FDEs. We prove the stability and convergence of the proposed methods. Numerical examples are given to support the theoretical results.

Analysis of two avian influenza epidemic models involving fractal-fractional derivatives with power and Mittag-Leffler memories.

Since certain species of domestic poultry and poultry are the main food source in many countries, the outbreak of avian influenza, such as H7N9, is a serious threat to the health and economy of those countries. This can be considered as the main reason for considering the preventive ways of avian influenza. In recent years, the disease has received worldwide attention, and a large variety of different mathematical models have been designed to investigate the dynamics of the avian influenza epidemic problem. In this paper, two fractional models with logistic growth and with incubation periods were considered using the Liouville-Caputo and the new definition of a nonlocal fractional derivative with the Mittag-Leffler kernel. Local stability of the equilibria of both models has been presented. For the Liouville-Caputo case, we have some special solutions using an iterative scheme via Laplace transform. Moreover, based on the trapezoidal product-integration rule, a novel iterative method is utilized to obtain approximate solutions for these models. In the Atangana-Baleanu-Caputo sense, we studied the uniqueness and existence of the solutions, and their corresponding numerical solutions were obtained using a novel numerical method. The method is based on the trapezoidal product-integration rule. Also, we consider fractal-fractional operators to capture self-similarities for both models. These novel operators predict chaotic behaviors involving the fractal derivative in convolution with power-law and the Mittag-Leffler function. These models were solved numerically via the Adams-Bashforth-Moulton and Adams-Moulton scheme, respectively. We have performed many numerical simulations to illustrate the analytical achievements. Numerical simulations show very high agreement between the acquired and the expected results.

An efficient numerical technique for a new fractional tuberculosis model with nonsingular derivative operator

The present study aims to investigate a new fractional model describing the dynamical behaviour of the tuberculosis infection. In this new formulation, we use a recently introduced fractional operator with Mittag–Leffler nonsingular kernel. To solve and simulate the proposed model, a new and efficient numerical method is developed based on the product-integration rule. Simulation results are provided and some discussions are given to verify the theoretical analysis. The results indicate that employing the nonsingular operator can extract the hidden aspects of the model under consideration while these features are invisible when we use the ordinary time-derivatives. Therefore, the non-integer calculus supplies more flexible models describing the asymptotic behaviours of the real-world phenomena and helps us to better understand their complex dynamics.

A mixed scheme of product integration rules in (−1,1)

The paper deals with the numerical approximation of integrals of the typeI(f,y):=∫−11f(x)k(x,y)dx,y∈S⊂R where f is a smooth function and the kernel k(x,y) involves some kinds of “pathologies” (for instance, weak singularities, high oscillations and/or endpoint algebraic singularities). We introduce and study a product integration rule obtained by interpolating f by an extended Lagrange polynomial based on Jacobi zeros. We prove that the rule is stable and convergent with the order of the best polynomial approximation of f in suitable function spaces. Moreover, we derive a general recurrence relation for the new modified moments appearing in the coefficients of the rule, just using the knowledge of the usual modified moments. The new quadrature sequence, suitable combined with the ordinary product rule, allows to obtain a “mixed” quadrature scheme, significantly reducing the number of involved samples of f. Numerical examples are provided in order to support the theoretical results and to show the efficiency of the procedure.

Existence and uniqueness of solution of a fractional order tuberculosis model

Tuberculosis is among the infectious diseases that kill human beings worldwide. This paper proposes a fractional order tuberculosis model that studies the dynamics of the disease. The operator considered here is the Atangana-Baleanu one in the Caputo sense. This is to include into the formulation of the model the effect of nonlocal fading memory. The existence and uniqueness of solution of the model is extensively studied. A numerical scheme is established based on the product-integration (PI) rule, which is used to solve the fractional model.

A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence.

The main objective of this research is to investigate a new fractional mathematical model involving a nonsingular derivative operator to discuss the clinical implications of diabetes and tuberculosis coexistence. The new model involves two distinct populations, diabetics and nondiabetics, while each of them consists of seven tuberculosis states: susceptible, fast and slow latent, actively tuberculosis infection, recovered, fast latent after reinfection, and drug-resistant. The fractional operator is also considered a recently introduced one with Mittag-Leffler nonsingular kernel. The basic properties of the new model including non-negative and bounded solution, invariant region, and equilibrium points are discussed thoroughly. To solve and simulate the proposed model, a new and efficient numerical method is established based on the product-integration rule. Numerical simulations are presented, and some discussions are given from the mathematical and biological viewpoints. Next, an optimal control problem is defined for the new model by introducing four control variables reducing the number of infected individuals. For the control problem, the necessary and sufficient conditions are derived and numerical simulations are given to verify the theoretical analysis.

Numerical solution of predator-prey model with Beddington-DeAngelis functional response and fractional derivatives with Mittag-Leffler kernel.

One of the major applications of the nonlinear system of differential equations in biomathematics is to describe the predator-prey problem. In this framework, the fractional predator-prey model with Beddington-DeAngelis is examined. This model is formed of three nonlinear ordinary differential equations to describe the interplay among populations of three species including prey, immature predator, and mature predator. The fractional operator used in this model is the Atangana-Baleanu fractional derivative in Caputo sense. We show first that the fractional predator-prey model has a unique solution, then propose an efficient numerical scheme based on the product integration rule. The numerical simulations indicate that the obtained approximate solutions are in excellent agreement with the expected theoretical results. The numerical method used in this paper can be utilized to solve other similar models.

A second-order accurate numerical method with graded meshes for an evolution equation with a weakly singular kernel

A second-order accurate numerical method with graded meshes is proposed and analyzed for an evolution equation with a weakly singular kernel. The graded meshes are employed to compensate for the singular behavior of the exact solution at t = 0 . For the time discretization, the product integration rule is used to approximate the Riemann–Liouville fractional integral, a generalized Crank–Nicolson time-stepping is considered and shown that the error is of order k 2 , where k denotes the maximum time step. A fully discrete difference scheme is constructed with space discretization by compact difference method. Numerical experiment is carried out to support the theoretical results. The comparison between the method on uniform grids and graded grids shows the efficiency of our method.

A product integration rule for hypersingular integrals on (0,+∞)

A product integration rule for hypersingular integrals on (0,+∞)

On the dense unbounded divergence of interpolatory product integration on Jacobi nodes

This paper deals with interpolatory product integration rules based on Jacobi nodes, associated with the Banach space of all s-times continuously differentiable functions, and with a Banach space of absolutely integrable functions, on the interval $$[-1,1]$$ of the real axis. In order to highlight the topological structure of the set of unbounded divergence for the corresponding product quadrature formulas, a family of continuous linear operators associated with these product integration procedures is pointed out, and the unboundedness of the set of their norms is established, by means of some properties involving the theory of Jacobi polynomials. The main result of the paper is based on some principles of Functional Analysis, and emphasizes the phenomenon of double condensation of singularities with respect to the considered interpolatory product quadrature formulas, by pointing out large subsets (in topological meaning) of the considered Banach spaces, on which the quadrature procedures are unboundedly divergent.

A fourth order product integration rule by using the generalized Euler–Maclaurin summation formula

The present paper deals with a variant of the generalized Euler–Maclaurin summation formula for the product integration rule. The main idea lies on introducing a mesh associated with the integral of the square root of the weight function. We construct the set of nodes and implement it for approximating the considered integrals. It is shown theoretically that the proposed quadrature rule is of fourth order. The results of the provided numerical examples confirm the theoretical prediction. Moreover, applications of the suggested scheme for approximating some real test problems such as weakly singular integrals, including a particular case of the Fermi–Dirac integral, are investigated.

A new quadrature scheme based on an Extended Lagrange Interpolation process

Let w(x)=e−xβxα, w¯(x)=xw(x) and let {pm(w)}m, {pm(w¯)}m be the corresponding sequences of orthonormal polynomials. Since the zeros of pm+1(w) interlace those of pm(w¯), it makes sense to construct an interpolation process essentially based on the zeros of Q2m+1:=pm+1(w)pm(w¯), which is called “Extended Lagrange Interpolation”. In this paper the convergence of this interpolation process is studied in suitable weighted L1 spaces, in a general framework which completes the results given by the same authors in weighted Lup((0,+∞)), 1≤p≤∞ (see [31], [28]). As an application of the theoretical results, an extended product integration rule, based on the aforesaid Lagrange process, is proposed in order to compute integrals of the type∫0+∞f(x)k(x,y)u(x)dx,u(x)=e−xβxγ(1+x)λ,γ>−1,λ∈R+, where the kernel k(x,y) can be of different kinds. The rule, which is stable and fast convergent, is used in order to construct a computational scheme involving the single product integration rule studied in [23]. It is shown that the “compound quadrature sequence” represents an efficient proposal for saving 1/3 of the evaluations of the function f, under unchanged speed of convergence.

On the topological structure of the set of singularities for interpolatory product integration rules

This paper highlights the phenomenon of double condensation of singularities for interpolatory product integration rules, related to a class of projection operators (including Lagrange and Sturm-Liouville projections), associated to the Banach space of real continuous functions defined on a compact set in RR of positive measure.

Product integration rules for Chebyshev weight functions with Chebyshev abscissae

We study two product integration rules, one for the Chebyshev weight of the first-kind based on the Chebyshev abscissae of the second-kind, and another one constructed the other way around, i.e., relative to the Chebyshev weight of the second-kind and based on the Chebyshev abscissae of the first-kind. The new rules are shown to have positive weights given by explicit formulae. Furthermore, we determine the precise degree of exactness and we compute the variance of the quadrature formulae, we examine their definiteness or nondefiniteness, and we obtain error bounds for these formulae either asymptotically optimal by Peano kernel methods or for analytic functions by Hilbert space techniques. In addition, the convergence of the quadrature formulae is shown not only for Riemann integrable functions on [−1,1], but also, by generalizing a result of Rabinowitz, for functions having a monotonic singularity at one or both endpoints of [−1,1].

Quadrature by expansion: A new method for the evaluation of layer potentials

Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and well-conditioned schemes in either interior or exterior domains. The practical application of these methods, however, requires the accurate evaluation of boundary integrals with singular, weakly singular or nearly singular kernels. Historically, these issues have been handled either by low-order product integration rules (computed semi-analytically), by singularity subtraction/cancellation, by kernel regularization and asymptotic analysis, or by the construction of special purpose “generalized Gaussian quadrature” rules. In this paper, we present a systematic, high-order approach that works for any singularity (including hypersingular kernels), based only on the assumption that the field induced by the integral operator is locally smooth when restricted to either the interior or the exterior. Discontinuities in the field across the boundary are permitted. The scheme, denoted QBX (quadrature by expansion), is easy to implement and compatible with fast hierarchical algorithms such as the fast multipole method. We include accuracy tests for a variety of integral operators in two dimensions on smooth and corner domains.

Numerical scheme for a scalar conservation law with memory

AbstractWe consider approximation of solutions to conservation laws with memory represented by a Volterra term with a smooth decreasing but possibly unbounded kernel. The numerical scheme combines Godunov method with a treatment of the integral term following from product integration rules. We prove stability for both linear and nonlinear flux functions and demonstrate the expected order of convergence using numerical experiments. The problem is motivated by modeling advective transport in heterogeneous media with subscale diffusion.Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 239–264, 2014

On accurate product integration rules for linear fractional differential equations

This paper addresses the numerical solution of linear fractional differential equations with a forcing term. Competitive and highly accurate Product Integration rules are derived by starting from an equivalent formulation in terms of a Volterra integral equation with a generalized Mittag-Leffler function in the kernel. The error analysis is reported and aspects related to the computational complexity are treated. Numerical tests confirming the theoretical findings are presented.

Orthogonality, interpolation and quadratures on the unit circle and the interval [formula omitted

Given a weight function σ ( x ) on [ − 1 , 1 ] , or more generally a positive Borel measure, the Erdős–Turán theorem assures the convergence in L 2 σ -norm to a function f of its sequence of interpolating polynomials at the zeros of the orthogonal polynomials or equivalently at the nodes of the Gauss–Christoffel quadrature formulas associated with σ . In this paper we will extend this result to the nodes of the Gauss–Radau and Gauss–Lobatto quadrature formulas by passing to the unit circle and taking advantage of the results on interpolation by means of Laurent polynomials at the zeros of certain para-orthogonal polynomials with respect to the weight function ω ( θ ) = σ ( cos θ ) | sin θ | on [ − π , π ] . As a consequence, an application to the construction of certain product integration rules on finite intervals of the real line will be given. Several numerical experiments are finally carried out.

Product integration methods based on discrete spline quasi-interpolants and application to weakly singular integral equations

Quadrature formulae are established for product integration rules based on discrete spline quasi-interpolants on a bounded interval. The integrand considered may have algebraic or logarithmic singularities. These formulae are then applied to the numerical solution of integral equations with weakly singular kernels.