Araştırma Makalesi
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Genocchi collocation method for accurate solution of nonlinear fractional differential equations with error analysis

Yıl 2023, Cilt: 3 Sayı: 4, 351 - 375, 30.12.2023

Mohamed El-gamel Nesreen Mohamed Waleed Adel

https://doi.org/10.53391/mmnsa.1373647
Cited By: 1

Öz

In this study, we introduce an innovative fractional Genocchi collocation method for solving nonlinear fractional differential equations, which have significant applications in science and engineering. The fractional derivative is defined in the Caputo sense and by leveraging fractional-order Genocchi polynomials, we transform the nonlinear problem into a system of nonlinear algebraic equations. A novel technique is employed to solve this system, enabling the determination of unknown coefficients and ultimately the solution. We derive the error bound for our proposed method and validate its efficacy through several test problems. Our results demonstrate superior accuracy compared to existing techniques in the literature, suggesting the potential for extending this approach to tackle more complex problems of critical physical significance.

Anahtar Kelimeler

Fractional-order modelling collocation method nonlinear phenomena error bound

Kaynakça

  • [1] Teodoro, G.S., Machado, J.T. and De Oliveira, E.C. A review of definitions of fractional derivatives and other operators. Journal of Computational Physics, 388, 195-208, (2019).
  • [2] Li, C., Qian, D. and Chen, Y.Q. On Riemann-Liouville and Caputo derivatives. Discrete Dynamics in Nature and Society, 2011, 562494, (2011).
  • [3] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. Theory and Applications of Fractional Differential Equations (Vol. 204). Elsevier: Netherlands, (2006).
  • [4] Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications (Vol. 198). Elsevier, (1999).
  • [5] Agarwal, R.P., Cuevas, C. and Soto, H. Pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Journal of Applied Mathematics and Computing, 37(1-2), 625-634, (2011).
  • [6] Ur Rahman, M., Arfan, M. and Baleanu, D. Piecewise fractional analysis of the migration effect in plant-pathogen-herbivore interactions. Bulletin of Biomathematics, 1(1), 1-23, (2023).
  • [7] Kurt, A., Tasbozan, O. and Durur, H. The exact solutions of conformable fractional partial differential equations using new sub equation method. Fundamental Journal of Mathematics and Applications, 2(2), 173-179, (2019).
  • [8] Yalçın Uzun, T. Oscillatory criteria of nonlinear higher order Ψ-Hilfer fractional differential equations. Fundamental Journal of Mathematics and Applications, 4(2), 134-142, (2021).
  • [9] Atede, A.O., Omame, A. and Inyama, S.C. A fractional order vaccination model for COVID-19 incorporating environmental transmission: a case study using Nigerian data. Bulletin of Biomathematics, 1(1), 78-110, (2023).
  • [10] Anjam, Y.N., Yavuz, M., Ur Rahman, M. and Batool, A. Analysis of a fractional pollution model in a system of three interconnecting lakes. AIMS Biophysics, 10(2), 220-240, (2023).
  • [11] Işık, E. and Daşbaşı, B. A compartmental fractional-order mobbing model and the determination of its parameters. Bulletin of Biomathematics, 1(2), 153-176, (2023).
  • [12] Yavuz, M., Sulaiman, T.A., Usta, F. and Bulut, H. Analysis and numerical computations of the fractional regularized long-wave equation with damping term. Mathematical Methods in the Applied Sciences, 44(9), 7538-7555, (2021).
  • [13] Yavuz, M., Özköse, F., Susam, M. and Kalidass, M. A new modeling of fractional-order and sensitivity analysis for Hepatitis-B disease with real data. Fractal and Fractional, 7(2), 165, (2023).
  • [14] Elsonbaty, A., Alharbi, M., El-Mesady, A. and Adel, W. Dynamical analysis of a novel discrete fractional lumpy skin disease model. Partial Differential Equations in Applied Mathematics, 9, 100604, (2024).
  • [15] El-Mesady, A., Adel, W., Elsadany, A.A. and Elsonbaty, A. Stability analysis and optimal control strategies of a fractional-order monkeypox virus infection model. Physica Scripta, 98(9), 095256, (2023).
  • [16] Evirgen, F., Uçar, E., Uçar, S. and Özdemir, N. Modelling influenza a disease dynamics under Caputo-Fabrizio fractional derivative with distinct contact rates. Mathematical Modelling and Numerical Simulation with Applications, 3(1), 58-73, (2023).
  • [17] Mpungu, K. and Ma’aruf Nass, A. On complete group classification of time fractional systems evolution differential equation with a constant delay. Fundamental Journal of Mathematics and Applications, 6(1), 12-23, (2023).
  • [18] Jajarmi, A. and Baleanu, D. A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems. Frontiers in Physics, 8, 220, (2020).
  • [19] Patnaik, S., Sidhardh, S. and Semperlotti, F. A Ritz-based finite element method for a fractional order boundary value problem of nonlocal elasticity. International Journal of Solids and Structures, 202, 398-417, (2020).
  • [20] Isah, A. and Phang, C. New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials, Journal of King Saud University-Science, 31(1), 1-7, (2019).
  • [21] El-Gamel, M. and El-Hady, M.A. Numerical solution of the Bagley-Torvik equation by Legendre-collocation method. SeMA Journal, 74, 371-383, (2017).
  • [22] Koundal, R., Kumar, R., Srivastava, K. and Baleanu, D. Lucas wavelet scheme for fractional Bagley–Torvik equations: Gauss–Jacobi approach. International Journal of Applied and Computational Mathematics, 8, 2-16, (2022).
  • [23] Abd-Elhameed, W.M. and Youssri, Y.H. Sixth-kind Chebyshev spectral approach for solving fractional differential equations. International Journal of Nonlinear Sciences and Numerical Simulation, 20(2), 191-203, (2019).
  • [24] Zaky, M.A. Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems. Applied Numerical Mathematics, 145, 429-457, (2019).
  • [25] Wang, C., Wang, Z. and Wang, L. A spectral collocation method for nonlinear fractional boundary value problems with a Caputo derivative. Journal of Scientific Computing, 76, 166- 188, (2018).
  • [26] Ismail, M., Saeed, U., Alzabut, J. and Ur Rehman, M. Approximate solutions for fractional boundary value problems via Green-CAS wavelet method. Mathematics, 7(12), 1164, (2019).
  • [27] Akgül, A. and Karatas Akgül, E. A novel method for solutions of fourth-order fractional boundary value problems, Fractal and Fractional, 3(2), 33, (2019).
  • [28] Li, X. and Wu, B. A new reproducing kernel collocation method for nonlocal fractional boundary value problems with non-smooth solutions. Applied Mathematics Letters, 86, 194-199, (2018).
  • [29] Ur Rehman, M. and Khan, R.A. A numerical method for solving boundary value problems for fractional differential equations. Applied Mathematical Modelling, 36(3), 894-907, (2012).
  • [30] Youssef, I.K. and El Dewaik, M.H. Solving Poisson’s equations with fractional order using Haar wavelet. Applied Mathematics and Nonlinear Sciences, 2(1), 271-284, (2017).
  • [31] Saeed, U. and Ur Rehman, M. Assessment of Haar wavelet-quasilinearization technique in heat convection-radiation equations. Applied Computational Intelligence and Soft Computing, 2014, 1–5, (2014).
  • [32] Pedas, A. and Tamme, E. Piecewise polynomial collocation for linear boundary value problems of fractional differential equations. Journal of Computational and Applied Mathematics, 236(13), 3349-3359, (2012).
  • [33] Pedas, A. and Tamme, E. Spline collocation for nonlinear fractional boundary value problems. Applied Mathematics and Computation, 244, 502-513, (2014).
  • [34] Ur Rehman, M. and Khan, R.A. The Legendre wavelet method for solving fractional differential equations. Communications in Nonlinear Science and Numerical Simulation, 16(11), 4163–4173, (2011).
  • [35] Araci, S. Novel identities for q-Genocchi numbers and polynomials. Journal of Function Spaces and Applications, 2012, 214961, (2012).
  • [36] Ozden, H., Simsek, Y. and Srivastava, H.M. A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Computers & Mathematics with Applications, 60(10), 2779–2787, (2010).
  • [37] Isah, A. and Phang, C. Operational matrix based on Genocchi polynomials for solution of delay differential equations. Ain Shams Engineering Journal, 9(4), 2123–2128, (2018).
  • [38] El-Gamel, M., Mohamed, N. and Adel, W. Numerical study of a nonlinear high order boundary value problems using Genocchi collocation technique. International Journal of Applied and Computational Mathematics, 8, 143, (2022).
  • [39] Li, Z., Yan, Y. and Ford, N.J. Error estimates of a high order numerical method for solving linear fractional differential equations. Applied Numerical Mathematics, 114, 201–220, (2017).
  • [40] Al-Mdallal, Q.M. and Hajji, M.A. A convergent algorithm for solving higher-order nonlinear fractional boundary value problems. Fractional Calculus and Applied Analysis, 18(6), 1423–1440, (2015).

Yıl 2023, Cilt: 3 Sayı: 4, 351 - 375, 30.12.2023

Mohamed El-gamel Nesreen Mohamed Waleed Adel

https://doi.org/10.53391/mmnsa.1373647
Cited By: 1

Öz

Kaynakça

  • [1] Teodoro, G.S., Machado, J.T. and De Oliveira, E.C. A review of definitions of fractional derivatives and other operators. Journal of Computational Physics, 388, 195-208, (2019).
  • [2] Li, C., Qian, D. and Chen, Y.Q. On Riemann-Liouville and Caputo derivatives. Discrete Dynamics in Nature and Society, 2011, 562494, (2011).
  • [3] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. Theory and Applications of Fractional Differential Equations (Vol. 204). Elsevier: Netherlands, (2006).
  • [4] Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications (Vol. 198). Elsevier, (1999).
  • [5] Agarwal, R.P., Cuevas, C. and Soto, H. Pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Journal of Applied Mathematics and Computing, 37(1-2), 625-634, (2011).
  • [6] Ur Rahman, M., Arfan, M. and Baleanu, D. Piecewise fractional analysis of the migration effect in plant-pathogen-herbivore interactions. Bulletin of Biomathematics, 1(1), 1-23, (2023).
  • [7] Kurt, A., Tasbozan, O. and Durur, H. The exact solutions of conformable fractional partial differential equations using new sub equation method. Fundamental Journal of Mathematics and Applications, 2(2), 173-179, (2019).
  • [8] Yalçın Uzun, T. Oscillatory criteria of nonlinear higher order Ψ-Hilfer fractional differential equations. Fundamental Journal of Mathematics and Applications, 4(2), 134-142, (2021).
  • [9] Atede, A.O., Omame, A. and Inyama, S.C. A fractional order vaccination model for COVID-19 incorporating environmental transmission: a case study using Nigerian data. Bulletin of Biomathematics, 1(1), 78-110, (2023).
  • [10] Anjam, Y.N., Yavuz, M., Ur Rahman, M. and Batool, A. Analysis of a fractional pollution model in a system of three interconnecting lakes. AIMS Biophysics, 10(2), 220-240, (2023).
  • [11] Işık, E. and Daşbaşı, B. A compartmental fractional-order mobbing model and the determination of its parameters. Bulletin of Biomathematics, 1(2), 153-176, (2023).
  • [12] Yavuz, M., Sulaiman, T.A., Usta, F. and Bulut, H. Analysis and numerical computations of the fractional regularized long-wave equation with damping term. Mathematical Methods in the Applied Sciences, 44(9), 7538-7555, (2021).
  • [13] Yavuz, M., Özköse, F., Susam, M. and Kalidass, M. A new modeling of fractional-order and sensitivity analysis for Hepatitis-B disease with real data. Fractal and Fractional, 7(2), 165, (2023).
  • [14] Elsonbaty, A., Alharbi, M., El-Mesady, A. and Adel, W. Dynamical analysis of a novel discrete fractional lumpy skin disease model. Partial Differential Equations in Applied Mathematics, 9, 100604, (2024).
  • [15] El-Mesady, A., Adel, W., Elsadany, A.A. and Elsonbaty, A. Stability analysis and optimal control strategies of a fractional-order monkeypox virus infection model. Physica Scripta, 98(9), 095256, (2023).
  • [16] Evirgen, F., Uçar, E., Uçar, S. and Özdemir, N. Modelling influenza a disease dynamics under Caputo-Fabrizio fractional derivative with distinct contact rates. Mathematical Modelling and Numerical Simulation with Applications, 3(1), 58-73, (2023).
  • [17] Mpungu, K. and Ma’aruf Nass, A. On complete group classification of time fractional systems evolution differential equation with a constant delay. Fundamental Journal of Mathematics and Applications, 6(1), 12-23, (2023).
  • [18] Jajarmi, A. and Baleanu, D. A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems. Frontiers in Physics, 8, 220, (2020).
  • [19] Patnaik, S., Sidhardh, S. and Semperlotti, F. A Ritz-based finite element method for a fractional order boundary value problem of nonlocal elasticity. International Journal of Solids and Structures, 202, 398-417, (2020).
  • [20] Isah, A. and Phang, C. New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials, Journal of King Saud University-Science, 31(1), 1-7, (2019).
  • [21] El-Gamel, M. and El-Hady, M.A. Numerical solution of the Bagley-Torvik equation by Legendre-collocation method. SeMA Journal, 74, 371-383, (2017).
  • [22] Koundal, R., Kumar, R., Srivastava, K. and Baleanu, D. Lucas wavelet scheme for fractional Bagley–Torvik equations: Gauss–Jacobi approach. International Journal of Applied and Computational Mathematics, 8, 2-16, (2022).
  • [23] Abd-Elhameed, W.M. and Youssri, Y.H. Sixth-kind Chebyshev spectral approach for solving fractional differential equations. International Journal of Nonlinear Sciences and Numerical Simulation, 20(2), 191-203, (2019).
  • [24] Zaky, M.A. Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems. Applied Numerical Mathematics, 145, 429-457, (2019).
  • [25] Wang, C., Wang, Z. and Wang, L. A spectral collocation method for nonlinear fractional boundary value problems with a Caputo derivative. Journal of Scientific Computing, 76, 166- 188, (2018).
  • [26] Ismail, M., Saeed, U., Alzabut, J. and Ur Rehman, M. Approximate solutions for fractional boundary value problems via Green-CAS wavelet method. Mathematics, 7(12), 1164, (2019).
  • [27] Akgül, A. and Karatas Akgül, E. A novel method for solutions of fourth-order fractional boundary value problems, Fractal and Fractional, 3(2), 33, (2019).
  • [28] Li, X. and Wu, B. A new reproducing kernel collocation method for nonlocal fractional boundary value problems with non-smooth solutions. Applied Mathematics Letters, 86, 194-199, (2018).
  • [29] Ur Rehman, M. and Khan, R.A. A numerical method for solving boundary value problems for fractional differential equations. Applied Mathematical Modelling, 36(3), 894-907, (2012).
  • [30] Youssef, I.K. and El Dewaik, M.H. Solving Poisson’s equations with fractional order using Haar wavelet. Applied Mathematics and Nonlinear Sciences, 2(1), 271-284, (2017).
  • [31] Saeed, U. and Ur Rehman, M. Assessment of Haar wavelet-quasilinearization technique in heat convection-radiation equations. Applied Computational Intelligence and Soft Computing, 2014, 1–5, (2014).
  • [32] Pedas, A. and Tamme, E. Piecewise polynomial collocation for linear boundary value problems of fractional differential equations. Journal of Computational and Applied Mathematics, 236(13), 3349-3359, (2012).
  • [33] Pedas, A. and Tamme, E. Spline collocation for nonlinear fractional boundary value problems. Applied Mathematics and Computation, 244, 502-513, (2014).
  • [34] Ur Rehman, M. and Khan, R.A. The Legendre wavelet method for solving fractional differential equations. Communications in Nonlinear Science and Numerical Simulation, 16(11), 4163–4173, (2011).
  • [35] Araci, S. Novel identities for q-Genocchi numbers and polynomials. Journal of Function Spaces and Applications, 2012, 214961, (2012).
  • [36] Ozden, H., Simsek, Y. and Srivastava, H.M. A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Computers & Mathematics with Applications, 60(10), 2779–2787, (2010).
  • [37] Isah, A. and Phang, C. Operational matrix based on Genocchi polynomials for solution of delay differential equations. Ain Shams Engineering Journal, 9(4), 2123–2128, (2018).
  • [38] El-Gamel, M., Mohamed, N. and Adel, W. Numerical study of a nonlinear high order boundary value problems using Genocchi collocation technique. International Journal of Applied and Computational Mathematics, 8, 143, (2022).
  • [39] Li, Z., Yan, Y. and Ford, N.J. Error estimates of a high order numerical method for solving linear fractional differential equations. Applied Numerical Mathematics, 114, 201–220, (2017).
  • [40] Al-Mdallal, Q.M. and Hajji, M.A. A convergent algorithm for solving higher-order nonlinear fractional boundary value problems. Fractional Calculus and Applied Analysis, 18(6), 1423–1440, (2015).
Toplam 40 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Sayısal Analiz, Uygulamalarda Dinamik Sistemler
Bölüm Araştırma Makalesi
Yazarlar

Mohamed El-gamel 0000-0003-2159-6619

Nesreen Mohamed Bu kişi benim 0000-0001-5143-1837

Waleed Adel 0000-0002-0557-8536

Yayımlanma Tarihi 30 Aralık 2023
Gönderilme Tarihi 9 Ekim 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 3 Sayı: 4

Kaynak Göster

APA El-gamel, M., Mohamed, N., & Adel, W. (2023). Genocchi collocation method for accurate solution of nonlinear fractional differential equations with error analysis. Mathematical Modelling and Numerical Simulation With Applications, 3(4), 351-375. https://doi.org/10.53391/mmnsa.1373647

Cited By

Spectral collocation with generalized Laguerre operational matrix for numerical solutions of fractional electrical circuit models
Mathematical Modelling and Numerical Simulation with Applications
https://doi.org/10.53391/mmnsa.1428035
 Kapak Resmi İndir


Math Model Numer Simul Appl - 2024 
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