Segmented Tau approximation for a forward–backward functional differential equation

Silva, Carmen Da; Escalante, René

doi:10.1016/j.camwa.2011.10.040

Cited by 6 publications

(8 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Developing numerical schemes for solving retarded, neutral, or mixed-type FDEs has attracted considerable attention in recent years and several methods have been proposed for them such as segmented LanczosTau Silva and Escalante (2011), forward/backward Euler (Ford and Lumb 2009;Ford et al 2010), trapezium rule Ford and Lumb (2009), Runge-Kutta (Wang 2016;Gao 2017), finite difference Mostaghim 2014, 2017), finite element Lima et al (2010), collocation , 2018Moghaddam and Aghili 2012;, and least square Teodoro et al (2009) methods. However, less attention has been paid to numerical techniques for solving mixed-type fractional-order FDEs (MFFDEs) as FDEs with retarded and neutral terms whose orders are extended to fractional numbers.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of mixed-type fractional functional differential equations using modified Lucas polynomials

et al. 2019

View full text Add to dashboard Cite

There is an increasing interest in the field of functional and fractional differential equations. The lack of closed-form analytical solutions motivates the development of numerical methods for solving mixed-type fractional-order functional differential equations (MFFDEs) with retarded and neutral terms. This paper studies the solution of MFFDEs by a collocation technique with modified Lucas polynomials. The proposed method uses operational matrices to obtain an approximate solution by means of a system of linear algebraic equations. The accuracy of the proposed algorithm is verified with three illustrative examples. Keyword Fractional calculus • Modified Lucas polynomials • Mixed-type differential equation • Neutral delay differential equations • Retarded delay differential equations • Spectral method Mathematics Subject Classification 34Axx • 34Kxx • 65M70 Communicated by Vasily E. Tarasov.

show abstract

Section: Introductionmentioning

confidence: 99%

Numerical solution of mixed-type fractional functional differential equations using modified Lucas polynomials

et al. 2019

View full text Add to dashboard Cite

show abstract

“…In [14], the step by step version of the Tau method was proposed by the authors to solve numerically the boundary value problem (1)-(3) in the autonomous case. There we observe that our numerical results were consistent with those reported by other authors using other numerical approaches (θmethod, least squares, and collocation methods, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…the above-mentioned methods), and where we obtained very satisfactory results. We think that, in a similar way as in [14] the segmented Tau method was applied to the autonomous case, it can also be extended to more complicated non-autonomous problem (1)- (3). By applying this method we seek to approximate the solution of equation ( 1) by a piecewise polynomial function [15].…”

Section: Introductionmentioning

confidence: 99%

“…In papers [39], [40], [42], and [41] the step by step Tau method was applied to find polynomial approximations to the solution of linear and nonlinear delay differential equations. Also recently in [14] the segmented Tau method was applied to find approximations to the solution of an autonomous mixed-type functional differential boundary value problem. These papers seem to show that the segmented Tau method is a natural and promising strategy in the numerical solution of functional differential equations.…”

Section: Introductionmentioning

confidence: 99%

“…For a brief exposition of the recursive formulation of the Tau method and the denominated canonical polynomials, the reader is refereed to [14], [39] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Segmented Tau Approximation for a Non-Autonomous Functional Differential Equation of Mixed Type

Silva¹,

Escalante²

2016

Preprint

Self Cite

View full text Add to dashboard Cite

The segmented formulation of the Tau method is used to numerically solve the non-autonomous forward-backward functional differential equationwhere x is the unknown function, a, b, and c are known functions. The step by step Tau method is applied to approximate the solution of this equation by a piecewise polynomial function. A boundary value problem is posed, numerically solved, and analyzed. Also, a novel way to generate a set of non-autonomous problems with known analytical solution is provided. From it, several nonautonomous problems were constructed and resolved with the proposed method. We conclude that the good numerical results obtained in our numerical experimentation and the relative simplicity of the Tau method demonstrate that it is a promising strategy for numerically solving mixed-type problems, as presented here.

show abstract

On the stability and behavior of solutions in mixed differential equations with delays and advances

Yeniçerioğlu

Yazıcı

Pinelas

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

In this study, some new results are obtained on asymptotic behavior and stability analysis of the mixed type differential equation where , and are real nonnegative continuous functions on , and and are real valued functions of bounded variation on . These results were obtained by using an appropriate real root of the characteristic equation. Moreover, by the use of two appropriate distinct real roots of the corresponding characteristic equation, a new result on the behavior of solutions is established. Five examples are also given to illustrate our results. We also presented the application of the obtained results in the special case of constant coefficients and have given three different cases in one example. The results obtained in this article show that real roots play an important role.

show abstract

Segmented Tau approximation for a forward–backward functional differential equation

Cited by 6 publications

References 31 publications

Numerical solution of mixed-type fractional functional differential equations using modified Lucas polynomials

Numerical solution of mixed-type fractional functional differential equations using modified Lucas polynomials

Segmented Tau Approximation for a Non-Autonomous Functional Differential Equation of Mixed Type

On the stability and behavior of solutions in mixed differential equations with delays and advances

Contact Info

Product

Resources

About