Distributed-order Fractional Derivative Research Articles

Advanced numerical simulation techniques in MHD fluid flow analysis using distributed fractional order derivatives and Cattaneo heat flux model

AbstractThis study investigates the flow and heat transfer characteristics of a second‐grade fluid over a flat surface with variable heat flux. Utilizing a mathematical model based on distributed order fractional derivatives, we offer a more precise representation of non‐Newtonian fluid behavior. The associated highly nonlinear equations are tackled numerically through an innovative amalgamation of the finite difference scheme and the technique. Our investigation thoroughly evaluates the influence of several critical parameters on the fluid's motion and thermal characteristics. Notably, the magnetic parameter significantly inhibits the fluid's velocity, illustrating the impact of magnetic forces. Additionally, the power law parameter leads to a decrease in both velocity and temperature profiles, highlighting its influence on fluid dynamics. Furthermore, changes in the Reynold number and the second‐grade fluid parameter are observed to cause a substantial increase in skin friction, by approximately 40% to 50%. These simulation results provide valuable insights into the complex interaction between fluid flow and heat transfer characteristics, enhancing our understanding of such systems in industrial applications.

A direct transcription method for solving distributed-order fractional optimal control problems

This paper considers the numerical solution of distributed-order Fractional Optimal Control Problems (FOCPs). The common approaches for solving these problems involve approximating the problem with a multi-term FOCP and subsequently discretizing the resulting multi-term problem using either a direct or indirect method. However, this paper bypasses the need for approximation with a multi-term FOCP by introducing innovative approximation formulas for distributed-order fractional derivatives. These approximation formulas are directly derived from the conventional Grünwald–Letnikov, L1, and trapezoidal formulas. Building upon these formulas, we develop the relevant fractional-order distributed-order operational matrices. Utilizing the operational matrix, we easily discretize the distributed-order FOCP to a Non-Linear Programming (NLP), which can be solved by a suitable NLP-solver. Additionally, to enhance the efficiency of solving the resulting NLP, we determine the closed-forms of both the Jacobian of constraints and the gradient of the objective function. The presented method is characterized by its speed, simplicity, and ease of implementation. Moreover, it can be used to solve a wide range of distributed-order FOCPs, such as those involving nonlinear dynamics, free final times, free terminal conditions, and path constraints. By means of ample numerical tests, the accuracy and efficiency of the proposed method are assessed.

Numerical study of distributed-order Bessel fractional derivative with application to Euler–Poisson–Darboux equation

The present paper introduces the distributed-order (DO) Bessel fractional derivative for study of the Euler–Poisson–Darboux (EPD) equation including the spatial Riesz fractional derivative (RFD). For this purpose, we discretize the integral term of the DO fractional derivative and approximate the RFD derivative. We thereafter apply an implicit difference method (IDM) for numerical analysis and solvability of the reduced system from the IDM is discussed. The stability and convergent theorems are stated and the numerical tests are given to show efficiency of the proposed technique and to verify good agreement with the theoretical concepts.

The distributed order models to characterize the flow and heat transfer of viscoelastic fluid between coaxial cylinders

The distributed order fractional derivatives can describe complex dynamic systems. In this paper, considering the periodic pressure gradient and magnetic field, the time distributed order fractional governing equations are established to simulate the two-dimensional flow and heat transfer of viscoelastic fluid between coaxial cylinders. Numerical solutions are obtained by the L1 approximation for the Caputo derivative (L1-scheme) and the finite difference method, and the effectiveness of numerical method is verified by a numerical example. Results demonstrate that the time distributed fractional Maxwell model can promote the flow while the distributed Cattaneo model can weaken heat transfer than the fractional Maxwell and Cattaneo model, and different weight coefficients have different effects on the fluid. The effect of physical parameters, such as the relaxation time of velocity and temperature λ 1, λ 2, the magnetic parameter M, the amplitude P 0 and frequency w of pressure gradient, and the Prandtl number Pr on velocity and temperature are discussed and analysed in detail.

An RBF-FD Method for Numerical Solutions of 2D Diffusion-Wave and Diffusion Equations of Distributed Fractional Order

AbstractThe subject of this paper is to propose a numerical algorithm for solving 2D diffusion and diffusion-wave equations of distributed order fractional derivatives. Such equations arise in modelling complex systems and have many important applications. Existence of integral term over the order of fractional derivative causes the high complexity of these equations and so their numerical solutions needs special cares. Using Gauss quadrature approach for discretizing the integral term of fractional derivative converts the distributed equation into a multi-term fractional differential equation. Then, the time variable is discretized with a suitable finite difference approach. The resultant semi-discretized equations are fully discretized by a radial basis function-generated finite difference based method. Convergence of the method are studied numerically. Various kind of test problems are considered for a comprehensive numerical study and the results confirm the efficiency of the method.

A computational approach for a system of coupled distributed-order fractional Klein–Gordon–Schrödinger equations

In this study, a system of coupled distributed-order fractional Klein–Gordon–Schrödinger equations is introduced. The distributed-order fractional derivative is generated based on the Caputo fractional differentiation. The discrete Chebyshev polynomials are handled to construct computational approach for this system. To do this, fractional derivatives matrices (distribute-order and classical) of the expressed discrete polynomials are obtained. The intended approach is based on approximating the imaginary and real parts of the system solution by the mentioned polynomials and applying the extracted operational matrices, along with employing the collocation technique. The constructed scheme transforms the solution of the original system into solving a related algebraic system. Three test problems are examined to confirm the adequacy of the expressed collocation procedure.

Numerical solution of distributed-order fractional 2D optimal control problems using the Bernstein polynomials

In this work, a new class of two-dimensional optimal control problems is introduced with the help of distributed-order fractional derivative in the Caputo form. The orthonormal Bernstein polynomials are used to make a numerical method to solve these problems. Through this way, some operational matrices are obtained for the classical and fractional derivatives of these polynomials to make effective utilisation of them in constructing the proposed method. The main advantage of the established method is that it turns the solution of the problem under consideration into a system of algebraic equations by approximating the state and control variables by the expressed polynomials and applying the method of Lagrange multipliers. The accuracy and capability of the proposed approach are investigated by solving some numerical examples.

Error analysis of a finite difference method for the distributed order sub-diffusion equation using discrete comparison principle

A finite difference method for numerically solving the initial boundary value problem of distributed order sub-diffusion equations with weakly singular solutions is presented, where Caputo fractional derivative is approximated by L1 scheme on nonuniform mesh and the space is discretized by finite difference method. The error analysis for the fully discrete scheme is obtained by using the discrete comparison principle. By carefully choosing a barrier function, the final error bound is β-robust which does not blow up when β→1−, where β (0<β<1) is considered to be the order of the distributed order fractional derivative operator. The numerical results are given and have shown that the error analysis is sharp.

Chelyshkov polynomials method for distributed-order time fractional nonlinear diffusion-wave equations

This work deals with the distributed-order time fractional nonlinear diffusion-wave equations. These equations are generated by replacing the first- and second-order time derivative terms with the distributed-order fractional derivative terms. The distributed-order fractional derivatives used in these problems are in the Caputo sense. The Chelyshkov polynomials as a well-known family of basis functions are used to develop a spectral collocation method for these problems. Through the way, some operational matrices regarding the classical and distribute-order fractional derivatives for these polynomials are extracted. In the proposed method, after approximating the solution of the problem in terms of the Chelyshkov polynomials and employing the expressed matrices, solving the primary equations transforms into solving algebraic systems which can be easily solved. Some numerical examples are studied to show the adequacy of the approach.

Numerical solution of distributed-order time fractional Klein–Gordon–Zakharov system

In this work, the distributed-order time fractional Klein–Gordon–Zakharov system is introduced by substituting the second-order temporal derivative with a distributed-order fractional derivative. The Caputo fractional derivative is utilized to define this kind of distributed-order fractional derivative. A high accuracy approach based on the Chebyshev cardinal polynomials is established for this system. The proposed method turns the fractional system solution into an algebraic system solution by approximating the unknown solution via these cardinal polynomials and engaging their derivative matrices (that are obtained in this paper). Some test problems are considered to investigate the capability and accuracy of this approach.

Wavelets based computational algorithms for multidimensional distributed order fractional differential equations with nonlinear source term

In this manuscript we provide effective computational algorithms based on Legendre wavelet (LW), Bernstein wavelet (BW), and standard tau approach to approximate the solution of viscoelasticity damping motion nonlinear distributed order fractional differential equation (VDMNDOFDE) and multi-dimensional distributed order time-fractional nonlinear partial differential equation (DOT-FNPDE). To the best of our understanding, the proposed computational algorithm is new and has not been previously applied for solving VDMNDOFDE and DOT-FNPDE. The matrix representation of distributed order fractional derivatives, integer order derivatives and product term associated with the integral based on LW and BW are established to find the numerical solutions of the proposed VDMNDOFDE and DOT-FNPDE. Moreover, the association of standard tau rule and Legendre-Gauss quadrature (LGQ) techniques along with constructed matrix representation of differential and integral operators diminish VDMNDOFDE and DOT-FNPDE into system of nonlinear algebraic equations. Here, these systems of nonlinear algebraic equations are solved by the Newton's method to get the value of unknown vectors. Error bounds, convergence analysis, numerical algorithms of the VDMNDOFDE and DOT-FNPDE are rigorously investigated. For the reliability of the proposed computational algorithm, numerous test examples have been incorporated in the manuscript to ensure the robustness and theoretical results of proposed technique. Also, the proposed technique found to be more accurate with the existing scheme.

Improved Gegenbauer spectral tau algorithms for distributed-order time-fractional telegraph models in multi-dimensions

The distributed-order fractional telegraph models are commonly used to describe the phenomenas of diffusion and wave-like anomalous, which can model processes without a power-law scale across the entire temporal domain. To increase the range of implementation of distributed-order fractional telegraph models, there is a need to present effective and accurate numerical algorithms to solve these models, as these models are hard to solve analytically. In this work, a novel matrix representation of the distributed-order fractional derivative based on shifted Gegenbauer (SG) polynomials has been derived. Also, two efficient algorithms based on the aforementioned operatonal matrix and the spectral tau method have been constructed for solving the one- and two-dimensional (1D and 2D) distributed-order time-fractional telegraph models with spatial variable coefficients. Also, a new operational matrix of the multiplication of space vectors has been built to have the ability in applying the tau method in the 2D case. The convergence and error bound analysis of the presented techniques are investigated. Moreover, the presented algorithms are applied on four miscellaneous test examples to illustrate the robustness and effectiveness of these algorithms.

A computational approach based on the fractional Euler functions and Chebyshev cardinal functions for distributed-order time fractional 2D diffusion equation

In this paper, the distributed-order time fractional diffusion equation is introduced and studied. The Caputo fractional derivative is utilized to define this distributed-order fractional derivative. A hybrid approach based on the fractional Euler functions and 2D Chebyshev cardinal functions is proposed to derive a numerical solution for the problem under consideration. It should be noted that the Chebyshev cardinal functions process many useful properties, such as orthogonality, cardinality and spectral accuracy. To construct the hybrid method, fractional derivative operational matrix of the fractional Euler functions and partial derivatives operational matrices of the 2D Chebyshev cardinal functions are obtained. Using the obtained operational matrices and the Gauss–Legendre quadrature formula as well as the collocation approach, an algebraic system of equations is derived instead of the main problem that can be solved easily. The accuracy of the approach is tested numerically by solving three examples. The reported results confirm that the established hybrid scheme is highly accurate in providing acceptable results.

Solving distributed-order fractional optimal control problems via the Fibonacci wavelet method

A new approach to finding the approximate solution of distributed-order fractional optimal control problems (D-O FOCPs) is proposed. This method is based on Fibonacci wavelets (FWs). We present a new Riemann–Liouville operational matrix for FWs using the hypergeometric function. Using this, an operational matrix of the distributed-order fractional derivative is presented. Implementing the mentioned operational matrix with the help of the Gauss–Legendre numerical integration, the problem converts to a system of algebraic equations. Error analysis is proposed. Finally, the validation of the present technique is checked by solving some numerical examples.

Recovery of a distributed order fractional derivative in an unknown medium

Recovery of a distributed order fractional derivative in an unknown medium

Novel and accurate Gegenbauer spectral tau algorithms for distributed order nonlinear time-fractional telegraph models in multi-dimensions

This paper sheds light on the numerical solutions of the multi-dimensional distributed-order (DO) nonlinear time-fractional telegraph equations (TFTEs) models. We present novel and accurate shifted Gegenbauer spectral tau schemes for solving the aforementioned models. In the proposed schemes, the known and unknown functions are approximated in terms of the shifted Gegenbauer polynomials (SGPs) by using the Kronecker product structure in time and space. The matrix representation of the DO fractional derivative of SGPs in the Caputo sense has been derived. Also, novel operational matrices of the multiplication of shifted Gegenbauer functions in multi-dimension have been constructed to remove the obstacle that facing the use of the tau method in the presence of nonlinear term or variable coefficients or both of them. The proposed method takes full advantage of the nonlocal nature of DO fractional differential operators. The presented method is applied on six miscellaneous test examples to illustrate its robustness and effectiveness for smooth and non smooth solutions.

Herglotz Variational Problems Involving Distributed-Order Fractional Derivatives with Arbitrary Smooth Kernels

In this paper, we consider Herglotz-type variational problems dealing with fractional derivatives of distributed-order with respect to another function. We prove necessary optimality conditions for the Herglotz fractional variational problem with and without time delay, with higher-order derivatives, and with several independent variables. Since the Herglotz-type variational problem is a generalization of the classical variational problem, our main results generalize several results from the fractional calculus of variations. To illustrate the theoretical developments included in this paper, we provide some examples.

Orthonormal piecewise Vieta-Lucas functions for the numerical solution of the one- and two-dimensional piecewise fractional Galilei invariant advection-diffusion equations

IntroductionRecently, a new family of fractional derivatives called the piecewise fractional derivatives has been introduced, arguing that for some problems, each of the classical fractional derivatives may not be able to provide an accurate statement of the consideration problem alone. In defining this kind of derivatives, several types of fractional derivatives can be used simultaneously. ObjectivesThis study introduces a new kind of piecewise fractional derivative by employing the Caputo type distributed-order fractional derivative and ABC fractional derivative. The one- and two-dimensional piecewise fractional Galilei invariant advection–diffusion equations are defined using this piecewise fractional derivative. MethodsA new class of basis functions called the orthonormal piecewise Vieta-Lucas (VL) functions are defined. Fractional derivatives of these functions in the Caputo and ABC senses are computed. These functions are utilized to construct two numerical methods for solving the introduced problems under non-local boundary conditions. The proposed methods convert solving the original problems into solving systems of algebraic equations. ResultsThe accuracy and convergence order of the proposed methods are examined by solving several examples. The obtained results are investigated, numerically. ConclusionThis study introduces a kind of piecewise fractional derivative. This derivative is employed to define the one- and two-dimensional piecewise fractional Galilei invariant advection–diffusion equations. Two numerical methods based on the orthonormal VL polynomials and orthonormal piecewise VL functions are established for these problems. The numerical results obtained from solving several examples confirm the high accuracy of the proposed methods.

Restrictions on parameters in distributed order fractional linear constitutive equations

We analyze a viscoelastic body in a linear stress state with the distributed order fractional derivative in the constitutive equation. The model was formulated so that it takes into account all derivatives of stress and strain between zero and one. Using a new procedure, we derive restrictions on a model that follow from the Clausius-Duhem inequality under isothermal conditions. Several constitutive equations are derived as special cases of proposed models. We apply our analysis to a creep test, a dynamical creep test and an oscillatory system.

A new operational matrix based on Müntz–Legendre polynomials for solving distributed order fractional differential equations

Our main aim in this work is to find the operational matrix of fractional derivative and the operational matrix of distributed order fractional derivative for the Müntz–Legendre polynomials (MLPs). The operational matrix approach with the tau method or collocation method is applied to reduce the solution of the linear/nonlinear distributed order fractional differential equations (DFDEs) to a system of linear/nonlinear algebraic equations. Moreover, seven numerical examples are included to show the validity and applicability of the suggested methods.