A new stable collocation method for solving a class of nonlinear fractional delay differential equations

Shi, Lei; Chen, Zhong; Ding, Xiaohua; Ma, Qiang

doi:10.1007/s11075-019-00858-9

Cited by 13 publications

(8 citation statements)

References 22 publications

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“…Shi et al. ( 2020 ); Nemati and Kalansara ( 2022 ) Consider the fractional Hutchinson-type model with variable coefficients subject to the terminal conditions for and . Here, the exact solution of this problem is or such that g ( t ) can be determined, respectively, as and When the problem is solved by the proposed method with and the residual error analysis technique with for both exact solution forms, such present results as the infinity normed error and timing complexity are compared in Tables 2 and 3 with those of the stable collocation method (SCM) Shi et al.…”

Section: Model Problemsmentioning

confidence: 99%

A streamlined numerical method to treat fractional nonlinear terminal value problems with multiple delays appearing in biomathematics

Kürkçü

2023

Comp. Appl. Math.

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In this study, a computational matrix-collocation method based on the Lagrange interpolation polynomial is specifically streamlined to treat the fractional nonlinear terminal value problems with multiple delays, such as the Hutchinson, the Wazewska-Czyzewska and the Lasota models in biomathematics. To do this, the robust nonlinear terms of which are smoothed to be deployed in the method. The uniqueness analysis of the solution is discussed in terms of the Banach contraction principle. An error analysis technique is non-linearly theorized and applied to improve the solutions. A programme for the method is especially developed. Thus, the outcomes of five fractional model problems constrained by terminal conditions are numerically and graphically evaluated in tables and figures. Based on the investigation of the results, one can claim that the method presents a sustainable and effective mathematical procedure for the aforementioned problems.

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Section: Model Problemsmentioning

confidence: 99%

A streamlined numerical method to treat fractional nonlinear terminal value problems with multiple delays appearing in biomathematics

Kürkçü

2023

Comp. Appl. Math.

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“…Stabilization problem of neutral FDDEs is given in [31]. Various numerical schemes [32,33,34,35] are designed by the researchers. Some issues related with the initialization of FDDEs are examined in [36].…”

Section: Introductionmentioning

confidence: 99%

Stability and Bifurcation Analysis of a Fractional Order Delay Differential Equation Involving Cubic Nonlinearity

Bhalekar,

Gupta

2022

Preprint

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Fractional derivative and delay are important tools in modeling memory properties in the natural system. This work deals with the stability analysis of a fractional order delay differential equationWe provide linearization of this system in a neighbourhood of equilibrium points and propose linearized stability conditions. To discuss the stability of equilibrium points, we propose various conditions on the parameters δ, , p, q and τ . Even though there are five parameters involved in the system, we are able to provide the stable region sketch in the qδ−plane for any positive and p. This provides the complete analysis of stability of the system. Further, we investigate chaos in the proposed model. This system exhibits chaos for a wide range of delay parameter.

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“…Therefore, different numerical methods are being introduced in getting the approximate solutions of different types of fractional differential equations. Some of the methods include variational iteration method, 11 homotopy perturbation method, 12 Adomian decomposition method, 13 collocation method, 14 Galerkin method, 15,16 and wavelets method. [17][18][19][20][21][22] In a very short period of time, wavelets has gained much more attention of many researchers due its vast applications in the fields of science and engineering.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Krawtchouk wavelets method for solving Caputo and Caputo–Hadamard fractional differential equations

Saeed

Idrees

Javid

et al. 2022

Math Methods in App Sciences

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The purpose of the present work is to develop a new wavelet method, named as Krawtchouk wavelets method, for solving both Caputo fractional and Caputo-Hadamard fractional differential equations on a semi-infinite domain.Design/methodology/approach: We have utilized the discrete Krawtchouk orthogonal polynomial for the construction of Krawtchouk wavelets method. The supporting analysis of the method such as construction of operational matrices, procedure of implementation, and convergence analysis of the method are being provided. We have also proposed a method by combining the Krawtchouk wavelets method with the method of step for the solution of Caputo-Hadamard fractional delay differential equations. Findings:We have provided the orthogonality condition for the Krawtchouk wavelets. We have derived and constructed the Krawtchouk wavelets matrix, Krawtchouk wavelets operational matrix of Riemann-Liouville and Hadamard-type fractional-order integration, and Krawtchouk wavelets operational matrix of Riemann-Liouville and Hadamard-type fractional-order integration for boundary value problems. These matrices are successfully utilized for the solution of Caputo and Caputo-Hadamard fractional differential equations. Operational matrices contains many zero entries, which makes the present method more efficient. Furthermore, we workout on the procedure of implementation of the method for the Caputo fractional differential equations as well as for the Caputo-Hadamard fractional differential equations. We also derived the convergence analysis of the Krawtchouk wavelets method, which completes the theoretical analysis of the proposed method.We have applied the Krawtchouk wavelets method for the numerical solutions of several Caputo fractional differential equations and Caputo-Hadamard fractional differential equations and compare the obtained results with the analytical solutions. The comparison shows the effectiveness of the present numerical method. In this paper, we have considered initial value problem, boundary value problem, and delay problem. According to the numerical results, the present method is more efficient and accurate.Originality/value: Since fractional differential equation is a latest and emerging field, many engineers and scientists can utilize the present method for solving their fractional models. To the best of the author's knowledge, the present

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A new stable collocation method for solving a class of nonlinear fractional delay differential equations

Cited by 13 publications

References 22 publications

A streamlined numerical method to treat fractional nonlinear terminal value problems with multiple delays appearing in biomathematics

A streamlined numerical method to treat fractional nonlinear terminal value problems with multiple delays appearing in biomathematics

Stability and Bifurcation Analysis of a Fractional Order Delay Differential Equation Involving Cubic Nonlinearity

Krawtchouk wavelets method for solving Caputo and Caputo–Hadamard fractional differential equations

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