Distributed-order Differential Equations Research Articles

Discretization and Stability Analysis for a Generalized Type Nonlinear Pharmacokinetic Models

Estimating the effects of drugs at different stages is directly proportional to the duration of recovery and the duration of pulling through with the disease. It is very important to estimate the effects of drugs at different stages. For this reason, solving Pharmacokinetic models which investigate these effects are very important. In this study, numerical solutions of one, two, and three-compartment nonlinear Pharmacokinetic models have been studied. Distributed order differential equations have been used for the solution. Numerical solutions have been found with the density function contained in distributed order differential equations and different values of this function. A nonstandard finite difference scheme has been used for numerical solutions. Finally, stability analyses of equilibrium points of the obtained discretized system have also been researched with the help of the Matignon criterion.

Numerical discretization and fast approximation of a variably distributed-order fractional wave equation

We investigate a variably distributed-order time-fractional wave partial differential equation, which could accurately model, e.g., the viscoelastic behavior in vibrations in complex surroundings with uncertainties or strong heterogeneity in the data. A standard composite rectangle formula of mesh size σ is firstly used to discretize the variably distributed-order integral and then the L-1 formula of degree of freedom N is applied for the resulting fractional derivatives. Optimal error estimates of the corresponding fully-discrete finite element method are proved based only on the smoothness assumptions of the data. To maintain the accuracy, setting σ = O(N−1) leads to O(N3) operations of evaluating the temporal discretization coefficients. To improve the computational efficiency, we develop a novel time-stepping scheme by expanding the fractional kernel at a fixed fractional order to decouple the fractional operator from the variably distributed-order integral. Only O(logN) terms are needed for the expansion without loss of accuracy, which consequently reduce the computational cost of generating coefficients from O(N3) to O(N2 logN). Optimal-order error estimates of this time-stepping scheme are rigorously proved via novel and different techniques from the standard analysis procedure of the L-1 methods. Numerical experiments are presented to substantiate the theoretical results.

REMARKS ON DECAYING SOLUTIONS TO DISTRIBUTED ORDER DIFFERENTIAL EQUATIONS

In this paper, we present results on the asymptotic behavior of the solutions to distributed order differential equations in Euclidean spaces. The paper develops some recent results in [1] on fractional differential equations to the case of more general nonlocal derivatives. We utilize the detailed behavior of the relaxation functions instead of the Mittag-Leffler functions. The typical decaying rate is logarithmic due to the ultra-slow phenomenon.

Haar wavelet method for solution of distributed order time-fractional differential equations

This manuscript is related to compute approximate solutions for a class of fractional distributed order differential equations (FDODEs). The corresponding derivative of fractional order is taken in Caputo sense. The adopted numerical scheme is based on collocation method together with Haar wavelet. This is an efficient numerical algorithm which convert the consider problem to a some system of equations of algebraic type. Upon utilizing the Broyden tools the obtain nonlinear system is solved for the intended numerical results. Further in case of linear equations the obtained system is solved by Gauss elimination procedure. To strengthen our results we testify several problems corresponding to different collocation points. We present also maximum absolute and root mean square errors. The graphical presentations are also given.

Finite difference/spectral methods for the two-dimensional distributed-order time-fractional cable equation

The cable equation plays a significant role in many areas of electrophysiology and in modeling neuronal dynamics. In recent years, considerable attention has been devoted to distributed-order differential equations because they appear to be more effective for modeling complex processes. In this work, a finite difference/Legendre spectral method is presented for the numerical simulation of the two-dimensional (2D) distributed-order time-fractional cable equation, where the finite difference method is employed in the temporal discretization and Legendre spectral method is adopted in the spatial discretization. The midpoint quadrature rule is used to approximate the distributed-order, such that the considered equation could be transformed into a multi-term fractional equation. The stability and convergence analysis of the proposed scheme is established, which illustrates that the numerical solution converges to the exact solution with order O(τ2+σ2+N−s), where τ, σ, N are the time step size, the step length in the approximation of the distributed-order and the polynomial degree, respectively. Furthermore, to demonstrate the versatility and applicability of our method, we provide numerical results that show good agreement with the theoretical analysis.

Two unconditionally stable difference schemes for time distributed-order differential equation based on Caputo\u2013Fabrizio fractional derivative

We consider distributed-order partial differential equations with time fractional derivative proposed by Caputo and Fabrizio in a one-dimensional space. Two finite difference schemes are established via Grünwald formula. We show that these two schemes are unconditionally stable with convergence rates O(tau ^{2}+h^{2}+ Delta alpha ^{2}) and O(tau ^{2}+h^{4}+Delta alpha ^{4}) in discrete L^{2}, respectively, where Δα, h, and τ are step sizes for distributed-order, space, and time variables, respectively. Finally, the performance of difference schemes is illustrated via numerical examples.

Fractional Backward Differential Formulas for the Distributed-Order Differential Equation with Time Delay

In this paper, we investigate the fractional backward differential formulas for the distributed-order differential equation with time delay. The mid-point quadrature rule is used to approximate the distributed-order equation by a multi-term fractional form. Next the transformed multi-term fractional equation is solved by discretizing in space by the fractional backward differential formulas method and in time by using the Crank–Nicolson scheme. We prove that proposed scheme is stable and convergent for $$\beta < \frac{5}{8}$$ with the accuracy $$\mathrm{O}({h^2} + {\kappa ^2} + {\sigma ^2})$$ . Finally, we give some examples to show the effectiveness of the numerical method and the results are in excellent agreement with the theoretical analysis.

Two alternating direction implicit spectral methods for two-dimensional distributed-order differential equation

In this paper, two alternating direction implicit Galerkin-Legendre spectral methods for distributed-order differential equation in two-dimensional space are developed. It is proved that the schemes are unconditionally stable and convergent with the convergence orders O(Δt + σ2 + N−m) and O(Δt2 + σ2 + N−m), respectively, where Δt, σ, N, and m are the time step size, step size in distributed-order variable, polynomial degree, and regularity in the space variable of the exact solution, respectively. Moreover, the applicability and accuracy of the two schemes are demonstrated by numerical experiments to support our theoretical analysis.

A block-centred finite difference method for the distributed-order differential equation with Neumann boundary condition

ABSTRACTIn this article, a block-centred finite difference method for the distributed-order differential equation with Neumann boundary condition is introduced and analysed. The stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence both for pressure and velocity are established on non-uniform rectangular grids, where and σ are the step sizes in time, space in x- and y-direction, and distributed order. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Analytic Solution of a System of Linear Distributed Order Differential Equations in the Reimann-Liouville Sense

In this paper, solution of a system of linear differential equations of distributed order in the Riemann-Liouville sense is analytically obtained. The distributed order relaxation equation is a special case of the system investigated in this paper. The solution of the mentioned system is introduced on the basis of a function which can be considered as the distributed order generalization of the matrix Mittag-Leffler functions. It is shown that this generalized function in two special cases of the weight function can be expressed in terms of the multivariate Mittag-Leffler functions and the Wright functions.

New compact difference scheme for solving the fourth-order time fractional sub-diffusion equation of the distributed order

In this paper, a class of new compact difference schemes is presented for solving the fourth-order time fractional sub-diffusion equation of the distributed order. By using an effective numerical quadrature rule based on boundary value method to discretize the integral term in the distributed-order derivative, the original distributed order differential equation is approximated by a multi-term time fractional sub-diffusion equation, which is then solved by a compact difference scheme. It is shown that the suggested compact difference scheme is stable and convergent in L∞ norm with the convergence order O(τ2+h4+(Δγ)p) when a boundary value method of order p is used, where τ,h and Δγ are the step sizes in time, space and distributed-order variables, respectively. Numerical results are reported to verify the high order accuracy and efficiency of the suggested scheme. Moreover, in the example, comparisons between some existing methods and the suggested scheme is also provided, showing that our method doesn't compromise in computational time.

A novel finite volume method for the Riesz space distributed-order diffusion equation

In recent years, considerable attention has been devoted to distributed-order differential equations mainly because they appear to be more effective for modelling complex processes which obey a mixture of power laws or flexible variations in space. In this paper, we propose a novel finite volume method (FVM) for a distributed-order space-fractional diffusion equation (FDE). Firstly, we use the mid-point quadrature rule to transform the space distributed-order diffusion equation into a multi-term fractional equation. Secondly, the transformed multi-term fractional equation is solved by discretising in space using the finite volume method and then in time using the Crank–Nicolson scheme. Thirdly, we prove that the Crank–Nicolson scheme with FVM is unconditionally stable and convergent with second order accuracy in both time and space. Finally, two numerical examples are presented to show the effectiveness of the numerical method. These methods and techniques can also be used to solve other types of fractional partial differential equations.

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...

A Petrov--Galerkin Spectral Method of Linear Complexity for Fractional Multiterm ODEs on the Half Line

We present a new tunably accurate Laguerre Petrov--Galerkin spectral method for solving linear multiterm fractional initial value problems with derivative orders at most one and constant coefficients on the half line. Our method results in a matrix equation of special structure which can be solved in $\mathcal{O}(N \log N)$ operations. We also take advantage of recurrence relations for the generalized associated Laguerre functions (GALFs) in order to derive explicit expressions for the entries of the stiffness and mass matrices, which can be factored into the product of a diagonal matrix and a lower-triangular Toeplitz matrix. The resulting spectral method is efficient for solving multiterm fractional differential equations with arbitrarily many terms, which we demonstrate by solving a fifty-term example. We apply this method to a distributed order differential equation, which is approximated by linear multiterm equations through the Gauss--Legendre quadrature rule. We provide numerical examples demonstra...

Chebyshev spectral approximation of distributed order differential equations

A Chebyshev collocation method is derived for the numerical approximation of distributed order differential equations in the Caputo sense. The numerical scheme is valid for both initial and boundary value problems. An error analysis is provided and illustrated through some numerical examples.

An implicit numerical method of a new time distributed-order and two-sided space-fractional advection-dispersion equation

Distributed-order differential equations have recently been investigated for complex dynamical systems, which have been used to describe some important physical phenomena. In this paper, a new time distributed-order and two-sided space-fractional advection-dispersion equation is considered. Firstly, we transform the time distributed-order fractional equation into a multi-term time-space fractional partial differential equation by applying numerical integration. Then an implicit numerical method is constructed to solve the multi-term fractional equation. The uniqueness, stability and convergence of the implicit numerical method are proved. Some numerical results are presented to demonstrate the effectiveness of the method. The method and techniques can be extended to other time distributed-order and space-fractional partial differential equations.

Some high-order difference schemes for the distributed-order differential equations

Two difference schemes are derived for both one-dimensional and two-dimensional distributed-order differential equations. It is proved that the schemes are unconditionally stable and convergent in an L1(L∞) norm with the convergence orders O(τ2+h2+Δα2) and O(τ2+h4+Δα4), respectively, where τ, h and Δα are the step sizes in time, space and distributed-order variables. Several numerical examples are given to confirm the theoretical results.

Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations

The Grünwald formula is used to solve the two-dimensional distributed-order differential equations. Two alternating direction implicit (ADI) difference schemes are derived. It is proved that the schemes are unconditionally stable and convergent with the convergence orders O(τ+h12+h22+Δα2) and O(τ+h14+h24+Δα4) in H1 norm, respectively, where τ,h1,h2 and Δα are the step sizes in time, space in x- and y-direction, and distributed order. The extrapolation method is applied to improve the approximate accuracy to the orders O(τ2∣lnτ∣2+h12+h22+Δα2) and O(τ2∣lnτ∣2+h14+h24+Δα4), respectively. Several numerical examples are given to confirm the theoretical results.

Matrix approach to discrete fractional calculus III: non-equidistant grids, variable step length and distributed orders

In this paper, we further develop Podlubny's matrix approach to discretization of integrals and derivatives of non-integer order. Numerical integration and differentiation on non-equidistant grids is introduced and illustrated by several examples of numerical solution of differential equations with fractional derivatives of constant orders and with distributed-order derivatives. In this paper, for the first time, we present a variable-step-length approach that we call 'the method of large steps', because it is applied in combination with the matrix approach for each 'large step'. This new method is also illustrated by an easy-to-follow example. The presented approach allows fractional-order and distributed-order differentiation and integration of non-uniformly sampled signals, and opens the way to development of variable- and adaptive-step-length techniques for fractional- and distributed-order differential equations.

Distributed order equations as boundary value problems

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.