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Kesirli Kısmi Diferansiyel Denklem Sistemlerinin Yaklaşık Analitik Çözümleri

Year 2017, Volume: 7 Issue: 1, 63 - 67, 01.01.2017

Ozan Özkan

Abstract

Bu makalede, kesirli kısmi diferansiyel denklem sistemlerinin çözümü için Kesirli Kompleks Diferansiyel Dönüşüm Yöntemi olarak adlandırılan yeni bir teknik önerilmektedir. Bu yöntem başlica iki yöntemin, yani Kesirli Kompleks Dönüşüm ve Diferansiyel Dönüşüm Yöntemi yöntemlerinin bir birleşimidir. Bu yeni yaklaşımın etkinliği, kesirli kısmi diferansiyel denklem sistemlerine uygulanarak başarıyla gösterilmiştir. Sonuç olarak, elde edilen sonuçlar bu yeni yaklaşımın çok etkili ve basit olduğunu ortaya koymaktadır

Keywords

Diferensiyel dönüşüm yöntemi Kesirli kompleks dönüşüm Kısmi türevli diferensiyel denklemler Kesirli kısmi diferensiyel denklemler

References

  • 1. Cansu, Ü. , Özkan, O. 2011. Solving Fokker-Planck Equation By Two-Dimensional Differential Transform, Adv. Math. Comput. Met., 1368,73
  • 2. Cassol M., Wortmann S., Rizza U. 2009. Analytic modeling of two-dimensional transient atmospheric pollutant dispersion by double GITT and Laplace Tran. techniques, E. Model. & Software, 24(1) 144-151.
  • 3. Chen, C. L. , Ho, S. H. 1999. Solving partial differential equations by two dimensional transform method, Appl. Math. Comput. 106, 171-179
  • 4. Cotta R. M., Mikhailov M. D. 1993. Integral transform method, Appl. Math. Model., 17 (3) 156-161.
  • 5. He, J. H., Elagan, S. K., Li, Z. B. 2012. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376, 4, pp. 257- -259.
  • 6. He, J. H. 2014. A Tutorial Review on Fractal Spacetime and Fractional Calculus, Int. J. Theoretical Phys., 53, 11, pp. 3698- 3718
  • 7. Hossein, J. and Daftardar-Gejji, V. 2006. “Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition”. Appl. Math. Comput 180.2: 488-497.
  • 8. Hossain, J. et al. 2012. “Solutions of the fractional DaveyStewartson equations with variational iteration method”. Rom. Rep. Phys 64.2: 337-346.
  • 9. Hossein, J. and Seifi S. 2009. “Solving a system of nonlinear fractional partial differential equations using homotopy analysis method”. Commun. Nonlinear Sci Numer Simul. 14.5: 1962-1969.
  • 10. Jang, M. J., Chen, C. L., Liu, Y. C. 2001. Two- dimensional differential transform for partial differential equations, Appl. Math. Comput. 121 , 261-270.
  • 11. Li ZB, He JH. 2010. Fractional Complex Transform for Fractional Differential Equations, Mat. Comput. Appl., 15 (5), 970-973.
  • 12. Li ZB. 2010. An Extended Fractional Complex Transform, J. Nonlinear Sci. Num. Simul., 11, 0335-0337.
  • 13. Mittag-Leffler G. M. 1903. “Une generalisation de l’intégrale de Laplace-Abel,” Comptes Rendus de l’Academie des Sciences Série II, vol. 137, pp. 537--539.
  • 14. Podlubny, I. 1999. Fractional Differential Equations, Academic Press, San Diego.
  • 15. Samko, S. G. , Kilbas, A. A. , and Marichev, O. I. 1993. Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon.
  • 16. Sejdic, E., Djurovic, I., Stankovic, L. J. 2011. Fractional Fourier transform as a signal processing tool: An overview of recent developments, Signal Process., 91(6) 1351-1369.
  • 17. Yang, X. 2011. Local Fractional Integral Transforms, Prog. Non. Sci., 4,1-225.
  • 18. Yanqin, L. 2012 . “Variational homotopy perturbation method for solving fractional initial boundary value problems. “ Abstr. Appl. Anal. Vol. 2012. Hindawi Publishing Corporation.
  • 19. Zhou, J. K. 1986. Differential Transformation and its Applications for Electric Circuits, Huazhong Univ. Press, Wuhan, China, in Chinese.
  • 20. Wazwaz, A. M. 2007. “The variational iteration method for solving linear and nonlinear systems of PDEs.” Comp. Math. App. 54.7: 895-902.

Approximate analytical solutions of systems of fractional partial differential equations

Year 2017, Volume: 7 Issue: 1, 63 - 67, 01.01.2017

Ozan Özkan

Abstract

In this article, a new technique which is called Fractional Complex Differential Transformation Method FCDTM is proposed for solving systems of fractional partial differential equations FPDEs . This method is mainly combination of two methods which are Fractional Complex Transform FCT and Differential Transform Method DTM . The efficiency of the new approach is illustrated by applying it successfully to systems of fractional partial differential equations. As a conclusion, the results reveal that the new approach is very effective and simple.

Keywords

Systems of fractional partial differential equations fractional complex transform differential transform method partial differential equation.

References

  • 1. Cansu, Ü. , Özkan, O. 2011. Solving Fokker-Planck Equation By Two-Dimensional Differential Transform, Adv. Math. Comput. Met., 1368,73
  • 2. Cassol M., Wortmann S., Rizza U. 2009. Analytic modeling of two-dimensional transient atmospheric pollutant dispersion by double GITT and Laplace Tran. techniques, E. Model. & Software, 24(1) 144-151.
  • 3. Chen, C. L. , Ho, S. H. 1999. Solving partial differential equations by two dimensional transform method, Appl. Math. Comput. 106, 171-179
  • 4. Cotta R. M., Mikhailov M. D. 1993. Integral transform method, Appl. Math. Model., 17 (3) 156-161.
  • 5. He, J. H., Elagan, S. K., Li, Z. B. 2012. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376, 4, pp. 257- -259.
  • 6. He, J. H. 2014. A Tutorial Review on Fractal Spacetime and Fractional Calculus, Int. J. Theoretical Phys., 53, 11, pp. 3698- 3718
  • 7. Hossein, J. and Daftardar-Gejji, V. 2006. “Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition”. Appl. Math. Comput 180.2: 488-497.
  • 8. Hossain, J. et al. 2012. “Solutions of the fractional DaveyStewartson equations with variational iteration method”. Rom. Rep. Phys 64.2: 337-346.
  • 9. Hossein, J. and Seifi S. 2009. “Solving a system of nonlinear fractional partial differential equations using homotopy analysis method”. Commun. Nonlinear Sci Numer Simul. 14.5: 1962-1969.
  • 10. Jang, M. J., Chen, C. L., Liu, Y. C. 2001. Two- dimensional differential transform for partial differential equations, Appl. Math. Comput. 121 , 261-270.
  • 11. Li ZB, He JH. 2010. Fractional Complex Transform for Fractional Differential Equations, Mat. Comput. Appl., 15 (5), 970-973.
  • 12. Li ZB. 2010. An Extended Fractional Complex Transform, J. Nonlinear Sci. Num. Simul., 11, 0335-0337.
  • 13. Mittag-Leffler G. M. 1903. “Une generalisation de l’intégrale de Laplace-Abel,” Comptes Rendus de l’Academie des Sciences Série II, vol. 137, pp. 537--539.
  • 14. Podlubny, I. 1999. Fractional Differential Equations, Academic Press, San Diego.
  • 15. Samko, S. G. , Kilbas, A. A. , and Marichev, O. I. 1993. Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon.
  • 16. Sejdic, E., Djurovic, I., Stankovic, L. J. 2011. Fractional Fourier transform as a signal processing tool: An overview of recent developments, Signal Process., 91(6) 1351-1369.
  • 17. Yang, X. 2011. Local Fractional Integral Transforms, Prog. Non. Sci., 4,1-225.
  • 18. Yanqin, L. 2012 . “Variational homotopy perturbation method for solving fractional initial boundary value problems. “ Abstr. Appl. Anal. Vol. 2012. Hindawi Publishing Corporation.
  • 19. Zhou, J. K. 1986. Differential Transformation and its Applications for Electric Circuits, Huazhong Univ. Press, Wuhan, China, in Chinese.
  • 20. Wazwaz, A. M. 2007. “The variational iteration method for solving linear and nonlinear systems of PDEs.” Comp. Math. App. 54.7: 895-902.
There are 20 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Ozan Özkan This is me

Publication Date January 1, 2017
Published in Issue Year 2017 Volume: 7 Issue: 1

Cite

APA Özkan, O. (2017). Approximate analytical solutions of systems of fractional partial differential equations. Karaelmas Fen Ve Mühendislik Dergisi, 7(1), 63-67.
AMA Özkan O. Approximate analytical solutions of systems of fractional partial differential equations. Karaelmas Fen ve Mühendislik Dergisi. January 2017;7(1):63-67.
Chicago Özkan, Ozan. “Approximate Analytical Solutions of Systems of Fractional Partial Differential Equations”. Karaelmas Fen Ve Mühendislik Dergisi 7, no. 1 (January 2017): 63-67.
EndNote Özkan O (January 1, 2017) Approximate analytical solutions of systems of fractional partial differential equations. Karaelmas Fen ve Mühendislik Dergisi 7 1 63–67.
IEEE O. Özkan, “Approximate analytical solutions of systems of fractional partial differential equations”, Karaelmas Fen ve Mühendislik Dergisi, vol. 7, no. 1, pp. 63–67, 2017.
ISNAD Özkan, Ozan. “Approximate Analytical Solutions of Systems of Fractional Partial Differential Equations”. Karaelmas Fen ve Mühendislik Dergisi 7/1 (January 2017), 63-67.
JAMA Özkan O. Approximate analytical solutions of systems of fractional partial differential equations. Karaelmas Fen ve Mühendislik Dergisi. 2017;7:63–67.
MLA Özkan, Ozan. “Approximate Analytical Solutions of Systems of Fractional Partial Differential Equations”. Karaelmas Fen Ve Mühendislik Dergisi, vol. 7, no. 1, 2017, pp. 63-67.
Vancouver Özkan O. Approximate analytical solutions of systems of fractional partial differential equations. Karaelmas Fen ve Mühendislik Dergisi. 2017;7(1):63-7.