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Conformable Kısmi Diferansiyel Denklemlerin Homotopi Analiz Yöntemi ile Nümerik Çözümleri

Year 2018, Volume: 18 Issue: 3, 842 - 851, 30.12.2018

Orkun Taşbozan , Gizem Bayaslı

Abstract

Bu makalede, kesirli Wu-Zhang sisteminin ve birleştirilmiş KdV-mKdV denklemidenkleminin nümerik çözümlerinielde etmek için Homotopi Analiz Yöntemi (HAM) uygulandı. Elde edilen sonuçlar, analitik çözümlerler ile karşılaştırıldı.

Keywords

Homotopi Analiz Yöntemi Conformable Kesirli Türev Kesirli Wu-Zhang Sistemi Kesirli Birleştirilmiş KdV-mKdV denklemi.

References

  • Abbasbandy S., 2006. The application of homotopy analysis method to nonlinear equations arising in heat transfer. Physics Letters A, 360, 109–113.
  • Abbasbandy S., Hashemi M.S. and Hashim I., 2013. On convergence of homotopy analysis method and its application to fractional integro-differential equations. Quaestiones Mathematicae, 36, 93-105.
  • Çenesiz Y., Baleanu D., Kurt A. and Tasbozan O., 2016. New exact solutions of Burgers’ type equations with conformable derivative. Waves in Random and Complex Media, 27, 103-116
  • Çenesiz Y., Tasbozan O. and Kurt A., 2017. Functional Variable Method for conformable fractional modified KdV-ZK equation and Maccari system. Tbilisi Mathematical Journal,10, 117-125.
  • Çenesiz Y., Tasbozan O. and Kurt A., 2017. On the New Solutions of the Conformable Time Fractional Generalized Hirota-Satsuma Coupled KdV System. Analele Universitatii de Vest, Timişoara Seria Matematica-Informatica LV, 55, 37- 49.
  • Debnath L. and Bhatta D., 2007. Integral Transforms and Their Applications, Chapman-Hall/CRC, USA. Esen A., Tasbozan O. and Yagmurlu N. M., 2012. Approximate Analytical Solutions of the Fractional Sharma-Tasso-Olver Equation Using Homotopy Analysis Method and a Comparison with Other Methods. Çankaya University Journal of Science and Engineering, 9, 139-147.
  • Esen A., Yagmurlu N. M. and Tasbozan O., 2013. Approximate Analytical Solution to Time-Fractional Damped Burger and Cahn-Allen Equations. Applied Mathematics & Information Sciences, 7, 1951-1956.
  • Eslami, M. and Rezazadeh, H., 2016. The first integral method forWu–Zhang system with conformable time-fractional derivative. Calcolo, 53, 475–485.
  • Eslami M. Rezazadeh H., Rezazadeh M. and Mosavi S.S., 2017. Exact solutions to the space–time fractional Schrödinger–Hirota equation and the space–time modified KDV- Zakharov–Kuznetsov equation. Optical and Quantum Electronics, 49, 279.
  • Hilfer P., 2000. Various Approaches to the Fractional CalculusApplications of Fractional Calculus In Physics. World Scientific, Germany. Hosseini K., Bekir A. and Ansari R., 2017. New exact solutions of the conformable time-fractionalCahn–Allen and Cahn–Hilliard equations using the modified Kudryashov method. Optik, 132, 203-209.
  • Hosseini K., Bejarbaneh E. Y., Bekir A. and Kaplan M., 2017. New exact solutions of some nonlinear evolution equations of pseudoparabolic type. Optical and Quantum Electronics, 49, 241.
  • Iyiola O.S., Tasbozan O., Kurt A. and Çenesiz Y., 2017. On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion. Chaos, Solitons and Fractals, 94, 1-7.
  • Kaplan M., 2017. Applications of two reliable methods for solving a nonlinear conformable time-fractional equation. Optical and Quantum Electronics, 49, 312.
  • Kaplan M., Bekir A. and Ozer M. N., 2017. A simple technique for constructing exact solutions to nonlinear differential equations with conformable fractional derivative. Optical and Quantum Electronics, 49, 266
  • . Khalil, R. and Horani, M.A., 2014. A new definition of fractional derivative. Journal of Computation and Applied Mathematics, 264, 65-70.
  • Khodadad F. S., Nazari F., Eslami M. and Rezazadeh H., 2017. Soliton solutions of the conformable fractional Zakharov–Kuznetsov equation with dual-power law Nonlinearity. Optical and Quantum Electronics, 49, 384.
  • Kilbas A.A., Srıvastava H.M. and Trujillo J.J., 2006. Theory and Applications of Fractional Differantial Equations, Elsevier, 0304-0208, vii s., New York.
  • Kumar D., Hosseini K. and Samadani F., 2017. The Sine-Gordon Expansion Method to Look For The Traveling Wave Solutions of The Tzitzeica Type Equations in Nonlinear Optics. Optik, 149, 439-446
  • Kurt, A., Çenesiz, Y. and Taşbozan, O., 2015. On the Solution of Burger’s equation with the new fractional derivative. Open Physics, 13, 355-360.
  • Kurt A., Tasbozan O. and Baleanu D., 2017. New solutions for conformable fractional Nizhnik–Novikov–Veselov system via 𝐺′/𝐺 expansion method and homotopy analysis methods. Optical and Quantum Electronics, 49, 333.
  • Kurt, A., Taşbozan, O. and Çenesiz, Y., 2016. Homotopy Analysis Method for Conformable Burgers-Korteweg-de Vries Equation. Bulletin of Mathematical Sciences and Applications,17,17-23
  • . Liao, S.J., 2003. Beyon Perturbation: Introduction to the Homotopy AnalysisMethod. CRC Press, Chapman&Hall, Boca Raton. Liao, S.J., 2009. Notes on the homotopy analysis method: Some definitions and theorems. Commun Nonlinear Sci Numer Simulat., 14, 983-997.
  • Miller, K.S. and Ross, B., 1993. An Introduction to The Fractional Calculus and Fractional Differantial Equations. J. Wiley-Sons, Canada. Molabahrami A. and Khani F., 2009. The homotopy analysis method to solve the Burgers-Huxley equation. Nonlinear Anal. B: Real World Appl., 10, 589-600.
  • Oldham K.B., Spainer J., 1974. The Fractional Calculus, Academic Press, New York. Podlubny, L., 1999. Fractional Differantial Equations. Academic Press, London.
  • Samko S.G., Kilbas A.A., Marichev O.I., 1993. Fractional Integrals and Derivative Theory and Applications, Gordon and Breach, 160 s, Longhorne. Taşbozan, O., Çenesiz, Y. and Kurt, A., 2016. New solutions for conformable fractional Boussinesq and combined KdV-mKdV equations using Jacobi elliptic function expansion method. The European Phypsical Journal Plus, 131, 244.
  • Taşbozan O., Esen A. and Yağmurlu N. M., 2012. Approximate Analytical Solutions of Fractional Coupled mKdV Equation by Homotopy Analysis Method. Open Journal of Applied Sciences, 2, 193-197.
  • Yavuz M., 2017. Novel solution methods for initial boundary value problems of fractional order with conformable differentiation. An International Journal of Optimization and Control: Theories & Applications, 8, 1-7.
  • Zhang X., Tang B. and He Y., 2011. Homotopy analysis method for higher-order fractional integro-differential equations. Computers and Mathematics with Applications,62, 3194-3203

Year 2018, Volume: 18 Issue: 3, 842 - 851, 30.12.2018

Orkun Taşbozan , Gizem Bayaslı

Abstract

References

  • Abbasbandy S., 2006. The application of homotopy analysis method to nonlinear equations arising in heat transfer. Physics Letters A, 360, 109–113.
  • Abbasbandy S., Hashemi M.S. and Hashim I., 2013. On convergence of homotopy analysis method and its application to fractional integro-differential equations. Quaestiones Mathematicae, 36, 93-105.
  • Çenesiz Y., Baleanu D., Kurt A. and Tasbozan O., 2016. New exact solutions of Burgers’ type equations with conformable derivative. Waves in Random and Complex Media, 27, 103-116
  • Çenesiz Y., Tasbozan O. and Kurt A., 2017. Functional Variable Method for conformable fractional modified KdV-ZK equation and Maccari system. Tbilisi Mathematical Journal,10, 117-125.
  • Çenesiz Y., Tasbozan O. and Kurt A., 2017. On the New Solutions of the Conformable Time Fractional Generalized Hirota-Satsuma Coupled KdV System. Analele Universitatii de Vest, Timişoara Seria Matematica-Informatica LV, 55, 37- 49.
  • Debnath L. and Bhatta D., 2007. Integral Transforms and Their Applications, Chapman-Hall/CRC, USA. Esen A., Tasbozan O. and Yagmurlu N. M., 2012. Approximate Analytical Solutions of the Fractional Sharma-Tasso-Olver Equation Using Homotopy Analysis Method and a Comparison with Other Methods. Çankaya University Journal of Science and Engineering, 9, 139-147.
  • Esen A., Yagmurlu N. M. and Tasbozan O., 2013. Approximate Analytical Solution to Time-Fractional Damped Burger and Cahn-Allen Equations. Applied Mathematics & Information Sciences, 7, 1951-1956.
  • Eslami, M. and Rezazadeh, H., 2016. The first integral method forWu–Zhang system with conformable time-fractional derivative. Calcolo, 53, 475–485.
  • Eslami M. Rezazadeh H., Rezazadeh M. and Mosavi S.S., 2017. Exact solutions to the space–time fractional Schrödinger–Hirota equation and the space–time modified KDV- Zakharov–Kuznetsov equation. Optical and Quantum Electronics, 49, 279.
  • Hilfer P., 2000. Various Approaches to the Fractional CalculusApplications of Fractional Calculus In Physics. World Scientific, Germany. Hosseini K., Bekir A. and Ansari R., 2017. New exact solutions of the conformable time-fractionalCahn–Allen and Cahn–Hilliard equations using the modified Kudryashov method. Optik, 132, 203-209.
  • Hosseini K., Bejarbaneh E. Y., Bekir A. and Kaplan M., 2017. New exact solutions of some nonlinear evolution equations of pseudoparabolic type. Optical and Quantum Electronics, 49, 241.
  • Iyiola O.S., Tasbozan O., Kurt A. and Çenesiz Y., 2017. On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion. Chaos, Solitons and Fractals, 94, 1-7.
  • Kaplan M., 2017. Applications of two reliable methods for solving a nonlinear conformable time-fractional equation. Optical and Quantum Electronics, 49, 312.
  • Kaplan M., Bekir A. and Ozer M. N., 2017. A simple technique for constructing exact solutions to nonlinear differential equations with conformable fractional derivative. Optical and Quantum Electronics, 49, 266
  • . Khalil, R. and Horani, M.A., 2014. A new definition of fractional derivative. Journal of Computation and Applied Mathematics, 264, 65-70.
  • Khodadad F. S., Nazari F., Eslami M. and Rezazadeh H., 2017. Soliton solutions of the conformable fractional Zakharov–Kuznetsov equation with dual-power law Nonlinearity. Optical and Quantum Electronics, 49, 384.
  • Kilbas A.A., Srıvastava H.M. and Trujillo J.J., 2006. Theory and Applications of Fractional Differantial Equations, Elsevier, 0304-0208, vii s., New York.
  • Kumar D., Hosseini K. and Samadani F., 2017. The Sine-Gordon Expansion Method to Look For The Traveling Wave Solutions of The Tzitzeica Type Equations in Nonlinear Optics. Optik, 149, 439-446
  • Kurt, A., Çenesiz, Y. and Taşbozan, O., 2015. On the Solution of Burger’s equation with the new fractional derivative. Open Physics, 13, 355-360.
  • Kurt A., Tasbozan O. and Baleanu D., 2017. New solutions for conformable fractional Nizhnik–Novikov–Veselov system via 𝐺′/𝐺 expansion method and homotopy analysis methods. Optical and Quantum Electronics, 49, 333.
  • Kurt, A., Taşbozan, O. and Çenesiz, Y., 2016. Homotopy Analysis Method for Conformable Burgers-Korteweg-de Vries Equation. Bulletin of Mathematical Sciences and Applications,17,17-23
  • . Liao, S.J., 2003. Beyon Perturbation: Introduction to the Homotopy AnalysisMethod. CRC Press, Chapman&Hall, Boca Raton. Liao, S.J., 2009. Notes on the homotopy analysis method: Some definitions and theorems. Commun Nonlinear Sci Numer Simulat., 14, 983-997.
  • Miller, K.S. and Ross, B., 1993. An Introduction to The Fractional Calculus and Fractional Differantial Equations. J. Wiley-Sons, Canada. Molabahrami A. and Khani F., 2009. The homotopy analysis method to solve the Burgers-Huxley equation. Nonlinear Anal. B: Real World Appl., 10, 589-600.
  • Oldham K.B., Spainer J., 1974. The Fractional Calculus, Academic Press, New York. Podlubny, L., 1999. Fractional Differantial Equations. Academic Press, London.
  • Samko S.G., Kilbas A.A., Marichev O.I., 1993. Fractional Integrals and Derivative Theory and Applications, Gordon and Breach, 160 s, Longhorne. Taşbozan, O., Çenesiz, Y. and Kurt, A., 2016. New solutions for conformable fractional Boussinesq and combined KdV-mKdV equations using Jacobi elliptic function expansion method. The European Phypsical Journal Plus, 131, 244.
  • Taşbozan O., Esen A. and Yağmurlu N. M., 2012. Approximate Analytical Solutions of Fractional Coupled mKdV Equation by Homotopy Analysis Method. Open Journal of Applied Sciences, 2, 193-197.
  • Yavuz M., 2017. Novel solution methods for initial boundary value problems of fractional order with conformable differentiation. An International Journal of Optimization and Control: Theories & Applications, 8, 1-7.
  • Zhang X., Tang B. and He Y., 2011. Homotopy analysis method for higher-order fractional integro-differential equations. Computers and Mathematics with Applications,62, 3194-3203
There are 28 citations in total.

Details

Primary Language Turkish
Journal Section Articles
Authors

Orkun Taşbozan

Gizem Bayaslı This is me

Publication Date December 30, 2018
Submission Date February 5, 2018
Published in Issue Year 2018 Volume: 18 Issue: 3

Cite

APA Taşbozan, O., & Bayaslı, G. (2018). Conformable Kısmi Diferansiyel Denklemlerin Homotopi Analiz Yöntemi ile Nümerik Çözümleri. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 18(3), 842-851.
AMA Taşbozan O, Bayaslı G. Conformable Kısmi Diferansiyel Denklemlerin Homotopi Analiz Yöntemi ile Nümerik Çözümleri. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. December 2018;18(3):842-851.
Chicago Taşbozan, Orkun, and Gizem Bayaslı. “Conformable Kısmi Diferansiyel Denklemlerin Homotopi Analiz Yöntemi Ile Nümerik Çözümleri”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 18, no. 3 (December 2018): 842-51.
EndNote Taşbozan O, Bayaslı G (December 1, 2018) Conformable Kısmi Diferansiyel Denklemlerin Homotopi Analiz Yöntemi ile Nümerik Çözümleri. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 18 3 842–851.
IEEE O. Taşbozan and G. Bayaslı, “Conformable Kısmi Diferansiyel Denklemlerin Homotopi Analiz Yöntemi ile Nümerik Çözümleri”, Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 18, no. 3, pp. 842–851, 2018.
ISNAD Taşbozan, Orkun - Bayaslı, Gizem. “Conformable Kısmi Diferansiyel Denklemlerin Homotopi Analiz Yöntemi Ile Nümerik Çözümleri”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 18/3 (December 2018), 842-851.
JAMA Taşbozan O, Bayaslı G. Conformable Kısmi Diferansiyel Denklemlerin Homotopi Analiz Yöntemi ile Nümerik Çözümleri. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2018;18:842–851.
MLA Taşbozan, Orkun and Gizem Bayaslı. “Conformable Kısmi Diferansiyel Denklemlerin Homotopi Analiz Yöntemi Ile Nümerik Çözümleri”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 18, no. 3, 2018, pp. 842-51.
Vancouver Taşbozan O, Bayaslı G. Conformable Kısmi Diferansiyel Denklemlerin Homotopi Analiz Yöntemi ile Nümerik Çözümleri. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2018;18(3):842-51.