scholarly journals A new approach for solving multi-pantograph type delay differential equations

Frede Nidal Anakira ◽  
Ali Jameel ◽  
Mohmmad Hijazi ◽  
Abdel-Kareem Alomari ◽  
Noraziah Man

<p>In this paper, a modified procedure based on the residual power series method (RPSM) was implemented to achieve approximate solution with high degree of accuracy for a system of multi-pantograph type delay differential equations (DDEs). This modified procedure is considered as a hybrid technique used to improve the curacy of the standard RPSM by combining the RPSM, Laplace transform and Pade approximant to be a powerful technique that can be solve the problems directly without large computational work, also even enlarge domain and leads to very accurate solutions or gives the exact solutions which is consider the best advantage of this technique. Some numerical applications are illustrated and numerical results are provided to prove the validity and the ability of this technique for this type of important differential equation that appears in different applications in engineering and control system.</p>

Mudaffer Alnobani ◽  
Omar Abu Al Yaqin

A new technique for solving a system of fractional Fredholm integro-differential equations (IDEs) is introduced in this manuscript. Furthermore, we present a review for the derivation of the residual power series method (RPSM) to solve fractional Fredholm IDEs in the paper done by Syam, as well as, corrections to the examples mentioned in that paper. The numerical results demonstrated the new technique’s applicability, efficacy, and high accuracy in dealing with these systems. On the other hand, a comparison has been done between the two schemes using the two corrected examples in addition to a problem that had been solved in many previous studies, and the results of these studies were compared with the new technique and RPSM. The comparison demonstrated clear superiority of our method over the RPSM for solving this class of equations. Moreover, they dispel the misconception that the RPSM works effectively on fractional Fredholm IDEs as mentioned in the paper done by Syam, whereas two problems solved by the RPSM produced an unaccepted error. Also, the comparison with the previous studies indicates the importance of the new method in dealing with the fractional Fredholm IDEs despite its simplicity, ease of use, and negligible computational time.

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1067-1085 ◽  
B.P. Mann ◽  
B.R. Patel

In this paper we describe a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second-order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems: systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous.

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Bochao Chen ◽  
Li Qin ◽  
Fei Xu ◽  
Jian Zu

This paper is devoted to studying the analytical series solutions for the differential equations with variable coefficients. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. These applications show that residual power series method is a simple, effective, and powerful method for seeking analytical series solutions of differential equations (especially for fractional differential equations) with variable coefficients.

2019 ◽  
Vol 11 (10) ◽  
pp. 168781401988103 ◽  
Asad Freihet ◽  
Shatha Hasan ◽  
Mohammad Alaroud ◽  
Mohammed Al-Smadi ◽  
Rokiah Rozita Ahmad ◽  

This article describes an efficient algorithm based on residual power series to approximate the solution of a class of partial differential equations of time-fractional Fokker–Planck model. The fractional derivative is assumed in the Caputo sense. The proposed algorithm gives the solution in a form of rapidly convergent fractional power series with easily computable coefficients. It does not require linearization, discretization, or small perturbation. To test simplicity, potentiality, and practical usefulness of the proposed algorithm, illustrative examples are provided. The approximate solutions of time-fractional Fokker–Planck equations are obtained by the residual power series method are compared with those obtained by other existing methods. The present results and graphics reveal the ability of residual power series method to deal with a wide range of partial fractional differential equations emerging in the modeling of physical phenomena of science and engineering.

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