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A collocation method to solve the parabolic-type partial integro-differential equations via Pell–Lucas polynomials

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  • Yüzbaşı, Şuayip
  • Yıldırım, Gamze

Abstract

In this paper, a new collocation method based on the Pell–Lucas polynomials is presented to solve the parabolic-type partial Volterra integro-differential equations. According to the method, it is assumed that the solution of this equation is in the formu2N(x,t)≅∑n=0N∑s=0Nan,sQn,s(x,t),Qn,s(x,t)=Qn(x)Qs(t)which depends on the Pell–Lucas polynomials. Next, the matrix representation of the solution is written. Using this matrix form, the matrix representations of the partial derivatives, the matrix representations of the Volterra integral part and the matrix forms of the conditions are also constituted. All obtained matrix forms are substituted in the equation and its conditions. Using equally spaced collocation points in matrix forms of this equation and initial conditions, the equation is reduced to a system of algebraic equations. The solution of this system gives the coefficients of the assumed solution. Additionally, the error analysis for the method is presented. According to this, an upper bound of the errors is determined. Also, the error estimation is made with the help of the residual function. Moreover, the residual improvement technique is also applied. Then, all these procedures are then supported with the examples. The results obtained from these examples are clearly tabulated and graphed. An important aspect of this study is to compare the obtained results with the present method with other results in the literature.

Suggested Citation

  • Yüzbaşı, Şuayip & Yıldırım, Gamze, 2022. "A collocation method to solve the parabolic-type partial integro-differential equations via Pell–Lucas polynomials," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    • Handle: RePEc:eee:apmaco:v:421:y:2022:i:c:s009630032200042x
    DOI: 10.1016/j.amc.2022.126956
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    References listed on IDEAS

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    1. Zhiyuan Li & Meichun Wang & Yulan Wang & Jing Pang, 2020. "Using Reproducing Kernel for Solving a Class of Fractional Order Integral Differential Equations," Advances in Mathematical Physics, Hindawi, vol. 2020, pages 1-12, March.
    2. M. Sameeh & A. Elsaid, 2016. "Chebyshev Collocation Method for Parabolic Partial Integrodifferential Equations," Advances in Mathematical Physics, Hindawi, vol. 2016, pages 1-7, December.
    3. Lu, Ziqiang & Zhu, Yuanguo, 2019. "Numerical approach for solution to an uncertain fractional differential equation," Applied Mathematics and Computation, Elsevier, vol. 343(C), pages 137-148.
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    5. Santanu Saha Ray & Rasajit K. Bera & Adem Kılıçman & Om P. Agrawal & Yasir Khan, 2015. "Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014," Abstract and Applied Analysis, Hindawi, vol. 2015, pages 1-2, March.
    6. Hajishafieiha, J. & Abbasbandy, S., 2020. "A new class of polynomial functions for approximate solution of generalized Benjamin–Bona–Mahony–Burgers (gBBMB) equations," Applied Mathematics and Computation, Elsevier, vol. 367(C).
    7. Al-Smadi, Mohammed & Arqub, Omar Abu, 2019. "Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 280-294.
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    Cited by:

    1. Deniz Elmaci & Nurcan Baykus & Savasaneril, 2022. "The Lucas Polynomial Solution Of Linear Volterra-Fredholm Integral Equations," Matrix Science Mathematic (MSMK), Zibeline International Publishing, vol. 6(1), pages 21-25, September.

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