Araştırma Makalesi
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Homotopi pertürbasyon Elzaki dönüşümü yöntemi ile doğrusal olmayan zaman-kesirli kısmi diferansiyel denklemler için yeni yaklaşık analitik çözümler

Yıl 2022, Cilt: 24 Sayı: 2, 468 - 482, 08.07.2022
https://doi.org/10.25092/baunfbed.984440

Öz

Bazı doğrusal olmayan zaman-kesirli mertebeden kısmi diferansiyel denklemler, homotopi pertürbasyon Elzaki dönüşümü yöntemi ile çözülmüştür. Kesirli türevler Caputo anlamında tanımlanmıştır. Uygulamalar homotopi pertürbasyon Elzaki dönüşümü yöntemi ile incelenmiştir. Bunun yanında, çözümlerin grafikleri MAPLE yazılımında çizdirilmiştir. Ayrıca homotopi pertürbasyon Elzaki dönüşümü yöntemi ve homotopi pertürbasyon Sumudu dönüşümü yöntemi çözümlerinin, lineer olmayan zaman-kesirli mertebeden kısmi diferansiyel denklemlerin tam çözümü ile mutlak hata karşılaştırması sunulmaktadır. Ek olarak, bu mutlak hata karşılaştırması tablolarda belirtilmiştir. Bu makalenin yeniliği, hem Caputo kesir dereceli gaz dinamiği denkleminin hem de Caputo kesir dereceli Klein-Gordon denkleminin bu yöntemle ilk analizidir. Bu nedenle, homotopi pertürbasyon Elzaki dönüşümü yöntemi, zaman-kesirli mertebeden kısmi diferansiyel denklemlerin analitik çözümlerinin elde edilmesinde hızlı ve etkilidir.

Kaynakça

  • Hilfer, R., Application of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, (2000).
  • Kilbas, A., Srivastava, H. and Trujillo, J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, (2006).
  • Miller, K. S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, (1993).
  • Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York, (1974).
  • Metzler, R. and Nonnenmacher, T. F., Space-and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation, Chemical Physics, 284, 1-2, 67-90, (2002).
  • Morgado, M. L. and Rebelo, M., Numerical approximation of distributed order reaction–diffusion equations, Journal of Computational and Applied Mathematics, 275, 216-227, (2015).
  • Baleanu, D., Jajarmi, A., Bonyah, E. and Hajipour, M., New aspects of poor nutrition in the life cycle within the fractional calculus, Advances in Difference Equations, 2018, 1, 1-14, (2018).
  • Jajarmi, A. and Baleanu, D., Suboptimal control of fractional-order dynamic systems with delay argument, Journal of Vibration and Control, 24, 12, 2430-2446, (2018).
  • Jajarmi, A. and Baleanu, D., A new fractional analysis on the interaction of HIV with CD4+ T-cells, Chaos, Solitons & Fractals, 113, 221-229, (2018).
  • Klimek, M., Fractional sequential mechanics-models with symmetric fractional derivative, Czechoslovak Journal of Physics, 51, 12, 1348-1354, (2001).
  • Laskin, N., Fractional quantum mechanics, Physical Review E, 62, 3, 3135-3145, (2000).
  • Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, (2010).
  • Wazwaz, A. M., A reliable modification of Adomian decomposition method, Applied Mathematics and Computation, 102, 1, 77-86, (1999).
  • He, J. H., Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation, 135, 1, 73-79, (2003).
  • He, J. H., Homotopy perturbation method for solving boundary value problems, Physics Letters, 350, 1-2, 87-88, (2006).
  • He, J. H., Addendum: new interpretation of homotopy perturbation method, International Journal of Modern Physics B, 20, 18, 2561-2568, (2006).
  • Yüzbaşı, Ş., A collocation method for numerical solutions of fractional-order logistic population model, International Journal of Biomathematics, 9, 02, 1650031, (2016).
  • Yüzbaşı, Ş., A numerical method for solving second-order linear partial differential equations under Dirichlet, Neumann and Robin boundary conditions, International Journal of Computational Methods, 14, 2, 1750015, (2017).
  • Yüzbaşı, Ş., A collocation approach for solving two-dimensional second-order linear hyperbolic equations, Applied Mathematics and Computation, 338, 101-114, (2018).
  • Merdan, M., Anaç, H. and Kesemen, T., The new Sumudu transform iterative method for studying the random component time-fractional Klein-Gordon equation, Sigma, 10, 3, 343-354, (2019).
  • Wang, K. and Liu, S., A new Sumudu transform iterative method for time-fractional Cauchy reaction-diffusion equation, Springer Plus, 5, 1, 865, (2016).
  • Anaç, H., Merdan, M., Bekiryazıcı, Z. and Kesemen, T., Bazı Rastgele Kısmi Diferansiyel Denklemlerin Diferansiyel Dönüşüm Metodu ve Laplace-Padé Metodu Kullanarak Çözümü, Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 9, 1, 108-118, (2019).
  • Ayaz, F., Solutions of the system of differential equations by differential transform method, Applied Mathematics and Computation, 147, 2, 547-567, (2004).
  • Kangalgil, F. and Ayaz, F., Solitary wave solutions for the KdV and mKdV equations by differential transform method, Chaos, Solitons & Fractals, 41, 1, 464-472, (2009).
  • Merdan, M., A new applicaiton of modified differential transformation method for modeling the pollution of a system of lakes, Selçuk Journal of Applied Mathematics, 11, 2, 27-40, (2010).
  • Zhou, J. K., Differential Transform and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan, (1986).
  • He, J. H., Variational iteration method-a kind of non-linear analytical technique: some examples, International Journal of Non-linear Mechanics, 34, 4, 699-708, (1999).
  • Elzaki, T. M., Applications of new transform “Elzaki transform” to partial differential equations, Global Journal of Pure and Applied Mathematics, 7, 1, 65-70, (2011).
  • Elzaki, T. M., Solution of nonlinear differential equations using mixture of Elzaki transform and differential transform method, In International Mathematical Forum, 7, 13, 631-638, (2012).
  • Elzaki, T. M. and Hilal, E. M. A., Homotopy perturbation and Elzaki transform for solving nonlinear partial differential equations, Mathematical Theory and Modeling, 2, 3, 33-42, (2012).
  • Elzaki, T. M. and Kim, H., The solution of radial diffusivity and shock wave equations by Elzaki variational iteration method, International Journal of Mathematical Analysis, 9, 22, 1065-1071, (2015).
  • Aggarwal, S., Chauhan, R. and Sharma, N., Application of Elzaki transform for solving linear Volterra integral equations of first kind, International Journal of Research in Advent Technology, 6, 12, 3687-3692, (2018).
  • Jena, R. M. and Chakraverty, S., Solving time-fractional Navier–Stokes equations using homotopy perturbation Elzaki transform, SN Applied Sciences, 1, 1, 1-16, (2019).
  • Akgül, E. K., Akgül, A. and Yavuz, M., New illustrative applications of integral transforms to financial models with different fractional derivatives, Chaos, Solitons & Fractals, 146, 110877, (2021).
  • Jena, R. M., Chakraverty, S., Yavuz, M. and Abdeljawad, T., A new modeling and existence–uniqueness analysis for Babesiosis disease of fractional order, Modern Physics Letters B, 35(30), 2150443, (2021).
  • Yavuz, M. and Sene, N., Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model, Journal of Ocean Engineering and Science, 6(2), 196-205, (2021).
  • Veeresha, P., A numerical approach to the coupled atmospheric ocean model using a fractional operator, Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(1), 1-10, (2021).
  • Yokuş, A., Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schrödinger equation, Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(1), 24-31, (2021).
  • Yavuz, M., Characterizations of two different fractional operators without singular kernel, Mathematical Modelling of Natural Phenomena, 14(3), 302, (2019).
  • Zada, L., Nawaz, R., Nisar, K. S., Tahir, M., Yavuz, M., Kaabar, M. K. and Martínez, F., New approximate-analytical solutions to partial differential equations via auxiliary function method, Partial Differential Equations in Applied Mathematics, 4, 100045, (2021).
  • Yavuz, M. and Sene, N., Approximate solutions of the model describing fluid flow using generalized ρ-laplace transform method and heat balance integral method, Axioms, 9(4), 123, (2020).
  • Yavuz, M. and Özdemir, N., Numerical inverse Laplace homotopy technique for fractional heat equations, Thermal Science, 22(1), 185-194, (2018).
  • Singh, J., Kumar, D. and Kılıçman, A., Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform, In Abstract and Applied Analysis, 2013, (2013), DOI: 10.1155/2013/934060.
  • Golmankhaneh, A. K., Golmankhaneh, A. K. and Baleanu, D., On nonlinear fractional Klein–Gordon equation, Signal Processing, 91, 3, 446-451, (2011).

New approximate-analytical solutions to nonlinear time-fractional partial differential equations via homotopy perturbation Elzaki transform method

Yıl 2022, Cilt: 24 Sayı: 2, 468 - 482, 08.07.2022
https://doi.org/10.25092/baunfbed.984440

Öz

Some nonlinear time-fractional partial differential equations are solved by homotopy perturbation Elzaki transform method. The fractional derivatives are defined in the Caputo sense. The applications are examined by homotopy perturbation Elzaki transform method. Besides, the graphs of the solutions are plotted in the MAPLE software. Also, absolute error comparison of homotopy perturbation Elzaki transform method and homotopy perturbation Sumudu transform method solutions with the exact solution of nonlinear time-fractional partial differential equations is presented. In addition, this absolute error comparison is indicated in the tables. The novelty of this article is the first analysis of both the gas dynamics equation of Caputo fractional order and the Klein-Gordon equation of Caputo fractional order via this method. Thus, homotopy perturbation Elzaki transform method is quick and effective in obtaining the analytical solutions of time-fractional partial differential equations.

Kaynakça

  • Hilfer, R., Application of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, (2000).
  • Kilbas, A., Srivastava, H. and Trujillo, J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, (2006).
  • Miller, K. S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, (1993).
  • Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York, (1974).
  • Metzler, R. and Nonnenmacher, T. F., Space-and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation, Chemical Physics, 284, 1-2, 67-90, (2002).
  • Morgado, M. L. and Rebelo, M., Numerical approximation of distributed order reaction–diffusion equations, Journal of Computational and Applied Mathematics, 275, 216-227, (2015).
  • Baleanu, D., Jajarmi, A., Bonyah, E. and Hajipour, M., New aspects of poor nutrition in the life cycle within the fractional calculus, Advances in Difference Equations, 2018, 1, 1-14, (2018).
  • Jajarmi, A. and Baleanu, D., Suboptimal control of fractional-order dynamic systems with delay argument, Journal of Vibration and Control, 24, 12, 2430-2446, (2018).
  • Jajarmi, A. and Baleanu, D., A new fractional analysis on the interaction of HIV with CD4+ T-cells, Chaos, Solitons & Fractals, 113, 221-229, (2018).
  • Klimek, M., Fractional sequential mechanics-models with symmetric fractional derivative, Czechoslovak Journal of Physics, 51, 12, 1348-1354, (2001).
  • Laskin, N., Fractional quantum mechanics, Physical Review E, 62, 3, 3135-3145, (2000).
  • Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, (2010).
  • Wazwaz, A. M., A reliable modification of Adomian decomposition method, Applied Mathematics and Computation, 102, 1, 77-86, (1999).
  • He, J. H., Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation, 135, 1, 73-79, (2003).
  • He, J. H., Homotopy perturbation method for solving boundary value problems, Physics Letters, 350, 1-2, 87-88, (2006).
  • He, J. H., Addendum: new interpretation of homotopy perturbation method, International Journal of Modern Physics B, 20, 18, 2561-2568, (2006).
  • Yüzbaşı, Ş., A collocation method for numerical solutions of fractional-order logistic population model, International Journal of Biomathematics, 9, 02, 1650031, (2016).
  • Yüzbaşı, Ş., A numerical method for solving second-order linear partial differential equations under Dirichlet, Neumann and Robin boundary conditions, International Journal of Computational Methods, 14, 2, 1750015, (2017).
  • Yüzbaşı, Ş., A collocation approach for solving two-dimensional second-order linear hyperbolic equations, Applied Mathematics and Computation, 338, 101-114, (2018).
  • Merdan, M., Anaç, H. and Kesemen, T., The new Sumudu transform iterative method for studying the random component time-fractional Klein-Gordon equation, Sigma, 10, 3, 343-354, (2019).
  • Wang, K. and Liu, S., A new Sumudu transform iterative method for time-fractional Cauchy reaction-diffusion equation, Springer Plus, 5, 1, 865, (2016).
  • Anaç, H., Merdan, M., Bekiryazıcı, Z. and Kesemen, T., Bazı Rastgele Kısmi Diferansiyel Denklemlerin Diferansiyel Dönüşüm Metodu ve Laplace-Padé Metodu Kullanarak Çözümü, Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 9, 1, 108-118, (2019).
  • Ayaz, F., Solutions of the system of differential equations by differential transform method, Applied Mathematics and Computation, 147, 2, 547-567, (2004).
  • Kangalgil, F. and Ayaz, F., Solitary wave solutions for the KdV and mKdV equations by differential transform method, Chaos, Solitons & Fractals, 41, 1, 464-472, (2009).
  • Merdan, M., A new applicaiton of modified differential transformation method for modeling the pollution of a system of lakes, Selçuk Journal of Applied Mathematics, 11, 2, 27-40, (2010).
  • Zhou, J. K., Differential Transform and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan, (1986).
  • He, J. H., Variational iteration method-a kind of non-linear analytical technique: some examples, International Journal of Non-linear Mechanics, 34, 4, 699-708, (1999).
  • Elzaki, T. M., Applications of new transform “Elzaki transform” to partial differential equations, Global Journal of Pure and Applied Mathematics, 7, 1, 65-70, (2011).
  • Elzaki, T. M., Solution of nonlinear differential equations using mixture of Elzaki transform and differential transform method, In International Mathematical Forum, 7, 13, 631-638, (2012).
  • Elzaki, T. M. and Hilal, E. M. A., Homotopy perturbation and Elzaki transform for solving nonlinear partial differential equations, Mathematical Theory and Modeling, 2, 3, 33-42, (2012).
  • Elzaki, T. M. and Kim, H., The solution of radial diffusivity and shock wave equations by Elzaki variational iteration method, International Journal of Mathematical Analysis, 9, 22, 1065-1071, (2015).
  • Aggarwal, S., Chauhan, R. and Sharma, N., Application of Elzaki transform for solving linear Volterra integral equations of first kind, International Journal of Research in Advent Technology, 6, 12, 3687-3692, (2018).
  • Jena, R. M. and Chakraverty, S., Solving time-fractional Navier–Stokes equations using homotopy perturbation Elzaki transform, SN Applied Sciences, 1, 1, 1-16, (2019).
  • Akgül, E. K., Akgül, A. and Yavuz, M., New illustrative applications of integral transforms to financial models with different fractional derivatives, Chaos, Solitons & Fractals, 146, 110877, (2021).
  • Jena, R. M., Chakraverty, S., Yavuz, M. and Abdeljawad, T., A new modeling and existence–uniqueness analysis for Babesiosis disease of fractional order, Modern Physics Letters B, 35(30), 2150443, (2021).
  • Yavuz, M. and Sene, N., Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model, Journal of Ocean Engineering and Science, 6(2), 196-205, (2021).
  • Veeresha, P., A numerical approach to the coupled atmospheric ocean model using a fractional operator, Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(1), 1-10, (2021).
  • Yokuş, A., Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schrödinger equation, Mathematical Modelling and Numerical Simulation with Applications (MMNSA), 1(1), 24-31, (2021).
  • Yavuz, M., Characterizations of two different fractional operators without singular kernel, Mathematical Modelling of Natural Phenomena, 14(3), 302, (2019).
  • Zada, L., Nawaz, R., Nisar, K. S., Tahir, M., Yavuz, M., Kaabar, M. K. and Martínez, F., New approximate-analytical solutions to partial differential equations via auxiliary function method, Partial Differential Equations in Applied Mathematics, 4, 100045, (2021).
  • Yavuz, M. and Sene, N., Approximate solutions of the model describing fluid flow using generalized ρ-laplace transform method and heat balance integral method, Axioms, 9(4), 123, (2020).
  • Yavuz, M. and Özdemir, N., Numerical inverse Laplace homotopy technique for fractional heat equations, Thermal Science, 22(1), 185-194, (2018).
  • Singh, J., Kumar, D. and Kılıçman, A., Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform, In Abstract and Applied Analysis, 2013, (2013), DOI: 10.1155/2013/934060.
  • Golmankhaneh, A. K., Golmankhaneh, A. K. and Baleanu, D., On nonlinear fractional Klein–Gordon equation, Signal Processing, 91, 3, 446-451, (2011).
Toplam 44 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Araştırma Makalesi
Yazarlar

Halil Anaç 0000-0002-1316-3947

Yayımlanma Tarihi 8 Temmuz 2022
Gönderilme Tarihi 18 Ağustos 2021
Yayımlandığı Sayı Yıl 2022 Cilt: 24 Sayı: 2

Kaynak Göster

APA Anaç, H. (2022). New approximate-analytical solutions to nonlinear time-fractional partial differential equations via homotopy perturbation Elzaki transform method. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 24(2), 468-482. https://doi.org/10.25092/baunfbed.984440
AMA Anaç H. New approximate-analytical solutions to nonlinear time-fractional partial differential equations via homotopy perturbation Elzaki transform method. BAUN Fen. Bil. Enst. Dergisi. Temmuz 2022;24(2):468-482. doi:10.25092/baunfbed.984440
Chicago Anaç, Halil. “New Approximate-Analytical Solutions to Nonlinear Time-Fractional Partial Differential Equations via Homotopy Perturbation Elzaki Transform Method”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24, sy. 2 (Temmuz 2022): 468-82. https://doi.org/10.25092/baunfbed.984440.
EndNote Anaç H (01 Temmuz 2022) New approximate-analytical solutions to nonlinear time-fractional partial differential equations via homotopy perturbation Elzaki transform method. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24 2 468–482.
IEEE H. Anaç, “New approximate-analytical solutions to nonlinear time-fractional partial differential equations via homotopy perturbation Elzaki transform method”, BAUN Fen. Bil. Enst. Dergisi, c. 24, sy. 2, ss. 468–482, 2022, doi: 10.25092/baunfbed.984440.
ISNAD Anaç, Halil. “New Approximate-Analytical Solutions to Nonlinear Time-Fractional Partial Differential Equations via Homotopy Perturbation Elzaki Transform Method”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 24/2 (Temmuz 2022), 468-482. https://doi.org/10.25092/baunfbed.984440.
JAMA Anaç H. New approximate-analytical solutions to nonlinear time-fractional partial differential equations via homotopy perturbation Elzaki transform method. BAUN Fen. Bil. Enst. Dergisi. 2022;24:468–482.
MLA Anaç, Halil. “New Approximate-Analytical Solutions to Nonlinear Time-Fractional Partial Differential Equations via Homotopy Perturbation Elzaki Transform Method”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, c. 24, sy. 2, 2022, ss. 468-82, doi:10.25092/baunfbed.984440.
Vancouver Anaç H. New approximate-analytical solutions to nonlinear time-fractional partial differential equations via homotopy perturbation Elzaki transform method. BAUN Fen. Bil. Enst. Dergisi. 2022;24(2):468-82.