IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v421y2022ics009630032200042x.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S009630032200042X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2022.126956?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    4. Polyanin, Andrei D., 2019. "Functional separable solutions of nonlinear reaction–diffusion equations with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 282-292.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wang, Jian & Zhu, Yuanguo & Gu, Yajing & Lu, Ziqiang, 2021. "Solutions of linear uncertain fractional order neutral differential equations," Applied Mathematics and Computation, Elsevier, vol. 407(C).
    2. Jin, Ting & Zhu, Yuanguo, 2020. "First hitting time about solution for an uncertain fractional differential equation and application to an uncertain risk index model," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    3. Jin, Ting & Yang, Xiangfeng, 2021. "Monotonicity theorem for the uncertain fractional differential equation and application to uncertain financial market," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 203-221.
    4. Jian Zhao & Zhenyue Chen & Jingqi Tu & Yunmei Zhao & Yiqun Dong, 2022. "Application of LSTM Approach for Predicting the Fission Swelling Behavior within a CERCER Composite Fuel," Energies, MDPI, vol. 15(23), pages 1-14, November.
    5. Hussam Aljarrah & Mohammad Alaroud & Anuar Ishak & Maslina Darus, 2022. "Approximate Solution of Nonlinear Time-Fractional PDEs by Laplace Residual Power Series Method," Mathematics, MDPI, vol. 10(12), pages 1-16, June.
    6. Majee, Suvankar & Jana, Soovoojeet & Das, Dhiraj Kumar & Kar, T.K., 2022. "Global dynamics of a fractional-order HFMD model incorporating optimal treatment and stochastic stability," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    7. Ngondiep, Eric, 2024. "A high-order combined finite element/interpolation approach for multidimensional nonlinear generalized Benjamin–Bona–Mahony–Burgers equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 215(C), pages 560-577.
    8. Lu, Qinyun & Zhu, Yuanguo, 2021. "LQ optimal control of fractional-order discrete-time uncertain systems," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    9. Liu, Yiyu & Zhu, Yuanguo & Lu, Ziqiang, 2021. "On Caputo-Hadamard uncertain fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    10. Andrei D. Polyanin & Vsevolod G. Sorokin, 2023. "Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays," Mathematics, MDPI, vol. 11(3), pages 1-25, January.
    11. Arqub, Omar Abu & Maayah, Banan, 2019. "Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC – Fractional Volterra integro-differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 394-402.
    12. Djennadi, Smina & Shawagfeh, Nabil & Abu Arqub, Omar, 2021. "A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    13. Hasan, Shatha & El-Ajou, Ahmad & Hadid, Samir & Al-Smadi, Mohammed & Momani, Shaher, 2020. "Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    14. Jin, Ting & Ding, Hui & Xia, Hongxuan & Bao, Jinfeng, 2021. "Reliability index and Asian barrier option pricing formulas of the uncertain fractional first-hitting time model with Caputo type," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    15. Qureshi, Sania & Memon, Zaib-un-Nisa, 2020. "Monotonically decreasing behavior of measles epidemic well captured by Atangana–Baleanu–Caputo fractional operator under real measles data of Pakistan," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    16. Andrei D. Polyanin, 2019. "Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions," Mathematics, MDPI, vol. 7(5), pages 1-19, April.
    17. Al-Smadi, Mohammed & Arqub, Omar Abu & Zeidan, Dia, 2021. "Fuzzy fractional differential equations under the Mittag-Leffler kernel differential operator of the ABC approach: Theorems and applications," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    18. Vsevolod G. Sorokin & Andrei V. Vyazmin, 2022. "Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration," Mathematics, MDPI, vol. 10(11), pages 1-39, May.
    19. Hussam Aljarrah & Mohammad Alaroud & Anuar Ishak & Maslina Darus, 2021. "Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations," Mathematics, MDPI, vol. 9(22), pages 1-17, November.
    20. Hasan, Shatha & Al-Smadi, Mohammed & El-Ajou, Ahmad & Momani, Shaher & Hadid, Samir & Al-Zhour, Zeyad, 2021. "Numerical approach in the Hilbert space to solve a fuzzy Atangana-Baleanu fractional hybrid system," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:421:y:2022:i:c:s009630032200042x. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.