A. Aminataei

Tau approximate solution of fractional partial differential equations

In this study, we improve the algebraic formulation of the fractional partial differential equations (FPDEs) by using the matrix-vector multiplication representation of the problem. This representation allows us to investigate an operational approach of the Tau method for the numerical solution of FPDEs. We introduce a converter matrix for the construction of converted Chebyshev and Legendre polynomials which is applied in the operational approach of the Tau method. We present the advantages of using the method and compare it with several other methods. Some experiments are applied to solve FPDEs including linear and nonlinear terms. By comparing the numerical results obtained from the other methods, we demonstrate the high accuracy and efficiency of the proposed method.

On the numerical solution of differential equations of Lane-Emden type

In this paper, a numerical method which produces an approximate polynomial solution is presented for solving Lane-Emden equations as singular initial value problems. Firstly, we use an integral operator (Yousefi (2006) [4]) and convert Lane-Emden equations into integral equations. Then, we convert the acquired integral equation into a power series. Finally, transforming the power series into Pade series form, we obtain an approximate polynomial of arbitrary order for solving Lane-Emden equations. The advantages of using the proposed method are presented. Then, an efficient error estimation for the proposed method is also introduced and finally some experiments and their numerical solutions are given; and comparing between the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method.

Operational Tau approximation for a general class of fractional integro-differential equations

In this work, an extension of the algebraic formulation of the operational Tau method (OTM) for the numerical solution of the linear and nonlinear fractional integro-differential equations (FIDEs) is proposed. The main idea behind the OTM is to convert the fractional differential and integral parts of the desired FIDE to some operational matrices. Then the FIDE reduces to a set of algebraic equations. We demonstrate the Tau matrix representation for solving FIDEs based on arbitrary orthogonal polynomials. Some advantages of using the method, errorestimation and computer algorithm are also presented. Illustrative linear and nonlinear experiments are included to show the validity and applicability of the presented method. Mathematical subject classification: 65M70, 34A25, 26A33, 47Gxx.

ON THE NUMERICAL SOLUTION OF NEUTRAL DELAY DIFFERENTIAL EQUATIONS USING MULTIQUADRIC APPROXIMATION SCHEME

Abstract. In this paper, the aim is to solve the neutral delay differentialequations in the following form using multiquadric approximation scheme,(1)ȷ y ′ ( t ) = f ( t,y ( t ) ,y ( t − τ ( t,y ( t ))) ,y ′ ( t − σ ( t,y ( t )))) , t 1 ≤ t ≤ t f ,y ( t ) = ϕ ( t ) , t ≤ t 1 , where f : [ t 1 ,t f ] × R × R × R → R is a smooth function, τ ( t,y ( t )) and σ ( t,y ( t )) are continuous functions on [ t 1 ,t f ] ×R such that t−τ ( t,y ( t )) <t f and t−σ ( t,y ( t )) < t f . Also ϕ ( t ) represents the initial function or theinitial data. Hence, we present the advantage of using the multiquadricapproximation scheme. In the sequel, presented numerical solutions ofsome experiments, illustrate the high accuracy and the efficiency of theproposed method even where the data points are scattered. 1. IntroductionNeutral delay differential equations (NDDEs) are considered as a branchof delay differential equations (DDEs). DDEs arise in many areas of variousmathematical modeling. For instance; infectious diseases, population dynam-ics, physiological and pharmaceutical kinetics and chemical kinetics, the navi-gational control of ships and aircrafts and control problems. There are manybooks to the application of DDEs which we can point out to the books of Driver[6], Gopalsamy [9], Halanay [11], Kolmanovskii and Myshkis [16], Kolmanovskiiand Nosov [17] and Kuang [18]. Some modelers ignore the ‘lag’ effect and usean ODE model as a substitute for a DDE model. Kuang ([18], p.11) commentsunder the heading

Multiquadric approximation scheme on the numerical solution of delay differential systems of neutral type

In this paper, the aim is to present the multiquadric approximation scheme on the numerical solution of delay differential systems of neutral type. In presenting the process of the solution, the error estimation and run time of the method is introduced. We present the advantages of using the method and compare it with other methods. Comparing the numerical results obtained from the other methods, demonstrate the high accuracy and the efficiency of the proposed method. Also, we present some experiments in which numerical results show that the method works excellently, even where the data points are scattered. This indicates that the method is stable too.

A numerical algorithm for the space and time fractional Fokker‐Planck equation

Purpose – The purpose of this paper is to present an algorithm based on operational Tau method (OTM) for solving fractional Fokker‐Planck equation (FFPE) with space‐ and time‐fractional derivatives. Fokker‐Planck equation with positive integer order is also considered.Design/methodology/approach – The proposed algorithm converts the desired FFPE to a set of algebraic equations using orthogonal polynomials as basis functions. The paper states some concepts, properties and advantages of proposed algorithm and its applications for solving FFPE.Findings – Some illustrative numerical experiments including linear and nonlinear FFPE are given and some comparisons are made between OTM and variational iteration method, Adomian decomposition method and homotpy perturbation method.Originality/value – Results demonstrate some capabilities of the proposed algorithm such as the simplicity, the accuracy and the convergency. Also, this is the first presentation of this algorithm for FFPE.

Numerical solution of differential algebraic equations using a multiquadric approximation scheme

The objective of this paper is to solve differential algebraic equations using a multiquadric approximation scheme. Therefore, we present the notation and basic definitions of the Hessenberg forms of the differential algebraic equations. In addition, we present the properties of the proposed multiquadric approximation scheme and its advantages, which include using data points in arbitrary locations with arbitrary ordering. Moreover, error estimation and the run time of the method are also considered. Finally some experiments were performed to illustrate the high accuracy and efficiency of the proposed method, even when the data points are scattered and have a closed metric.

An algorithm for solving multi-term diffusion-wave equations of fractional order

In this paper an algorithm, based on a new modified homotopy perturbation method (MHPM), is presented to obtain approximate solutions of multi-term diffusion-wave equations of fractional order. To illustrate the method some examples are provided. The results show the simplicity and the efficiency of the algorithm.

Quadric spline approximation scheme for solving retarded delay differential equations

Abstract In this paper, the objective is to solve certain retarded delay differential equations using a quadric spline approximation scheme. The advantage of using the proposed method is by numerical experiment that demonstrates the high accuracy and efficiency of themethod even where the data points are scattered.