The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method

Ali Kurt; Orkun Tasbozan; Hulya Durur

doi:10.33401/fujma.562819

Araştırma Makalesi

Yıl 2019, Cilt: 2 Sayı: 2, 173 - 179, 20.12.2019

Ali Kurt Orkun Tasbozan Hulya Durur

https://doi.org/10.33401/fujma.562819

Cited By: 10

Öz

Kaynakça

[1] K. Oldham, J. Spanier, The Fractional Calculus, Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, 1974.
[2] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, 1993.
[3] I. Podlubny, Fractional Differential Equations, Academic Press,1999.
[4] A. A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
[5] A. Kurt, O. Tasbozan, Approximate analytical solution of the time fractional Whitham-Broer-Kaup equation using the homotopy analysis method, Int. J. Pure Appl. Math., 98(4) (2015), 503-510.
[6] O. Tasbozan, A. Esen, N. M. Yagmurlu, Y. Ucar, A numerical solution to fractional diffusion equation for force-free case, Abstr. Appl. Anal., 2013, Hindawi, (2013).
[7] C. Celik, M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. of Comput. Phys., 231(4) (2012), 1743-1750.
[8] Y. Cenesiz, A. Kurt, New fractional complex transform for conformable fractional partial differential equations, J. Appl. Math. Stat. Inf., 12(2) (2016), 41-47.
[9] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.
[10] M. Eslami, H. Rezazadeh, The first integral method for Wu-Zhang system with conformable time-fractional derivative, Calcolo, 53(3) (2016), 475-485.
[11] H. Aminikhah, A. R. Sheikhani, H. Rezazadeh, Sub-equation method for the fractional regularized long-wave equations with conformable fractional derivatives, Sci. Iran. Transaction B, Mech. Eng., 23(3) (2016), 1048.
[12] M. S. Osman, A. Korkmaz, H. Rezazadeh, M. Mirzazadeh, M. Eslami, Q. Zhou, The unified method for conformable time fractional Schrdinger equation with perturbation terms, Chinese J. Phys., 56(5) (2018), 2500-2506.
[13] Y. Cenesiz, D. Baleanu, A. Kurt, O. Tasbozan, New exact solutions of Burgers’ type equations with conformable derivative, Wave. Random. Complex, 27(1) (2017), 103-116.
[14] A. Kurt, O. Tasbozan, D. Baleanu, New solutions for conformable fractional Nizhnik-Novikov-Veselov system via G0=G expansion method and homotopy analysis methods, Opt. Quant. Electron., 49(10) (2017), 333.
[15] K. Hosseini, P. Mayeli, R. Ansari, Bright and singular soliton solutions of the conformable time-fractional Klein-Gordon equations with different nonlinearities, Wave. Random Complex , 28(3) (2018), 426-434.
[16] A. Korkmaz, K. Hosseini, Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods, Opt. Quant. Electron., 49(8) (2017), 278.
[17] H. Rezazadeh, H. Tariq, M. Eslami, M. Mirzazadeh, Q. Zhou, New exact solutions of nonlinear conformable time-fractional Phi-4 equation, Chinese J. Phys., 56(6) (2018), 2805-2816.
[18] H. Bulut, T.A. Sulaiman, H.M. Baskonus, H. Rezazadeh, M. Eslami, M. Mirzazadeh, Optical solitons and other solutions to the conformable space-time fractional Fokas-Lenells equation, Optik, 172 (2018), 20-27.
[19] H. Rezazadeh, S. M. Mirhosseini-Alizamini, M. Eslami, M. Rezazadeh, M. Mirzazadeh, S. Abbagari, New optical solitons of nonlinear conformable fractional Schr¨odinger-Hirota equation, Optik, 172 (2018), 545-553.
[20] I. E. Inan, Multiple soliton solutions of some nonlinear partial differential equations, Univers. J. Math. Appl., 1(4) (2018), 273-279.
[21] H. Rezazadeh, M. S. Osman, M. Eslami, M. Ekici, A. Sonmezoglu, M. Asma, W. A. M. Othman, B. R. Wong, M. Mirzazadeh, Q. Zhou, A. Biswas, M. Belic, Mitigating Internet bottleneck with fractional temporal evolution of optical solitons having quadratic-cubic nonlinearity, Optik, 164 (2018), 84-92.
[22] A. Biswas, M. O. Al-Amr, H. Rezazadeh, M. Mirzazadeh, M. Eslami, Q. Zhou, S. P. Moshokoa, M. Belic, Resonant optical solitons with dual-power law nonlinearity and fractional temporal evolution, Optik, 165 (2018), 233-239.
[23] H. Bulut, T. A. Sulaiman, H. M. Baskonus, Dark, bright optical and other solitons with conformable space-time fractional second-order spatiotemporal dispersion, Optik, 163 (2018), 1-7.
[24] M. H. Cherif, D. Ziane, Homotopy analysis Aboodh transform method for nonlinear system of partial differential Equations, Univers. J. Math. Appl., 1(4) (2018), 244-253.
[25] A. M. Wazwaz, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, App. Math. Comput., 184(2) (2007), 1002-1014.
[26] D. Ziane, T. M. Elzaki, M. Hamdi Cherif, Elzaki transform combined with variational iteration method for partial differential equations of fractional order, Fundam. J. Math. Appl., 1(1) (2018), 102-108.
[27] D. Feng, K. Li, On exact traveling wave solutions for (1+ 1) dimensional Kaup-Kupershmidt equation, Appl. Math., 2(6) (2011), 752-756.
[28] C. A. Gomez S, New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations, Appl. Math. Comput., 216(1) (2010), 241-250.
[29] F. Tascan, A. Akbulut, Construction of exact solutions to partial differential equations with CRE method, Commun. Adv. Math. Sci., 2(2) (2019), 105-113.
[30] M. H. Cherif, D. Ziane, Variational iteration method combined with new transform to solve fractional partial differential equations, Univers. J. Math. Appl., 1(2) (2018), 113-120.
[31] A. H. Salas, Solving the generalized Kaup-Kupershmidt equation, Adv. Studies Theor. Phys., 6(18) (2012), 879-885.
[32] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.
[33] H. Rezazadeh, A. Korkmaz, M. Eslami, J. Vahidi, R. Asghari, Traveling wave solution of conformable fractional generalized reaction Duffing model by generalized projective Riccati equation method, Opt. Quant. Electron., 50(3) (2018), 150.
[34] R. Polat, Finite difference solution to the space-time fractional partial differential-difference Toda lattice equation, J. Math. Sci. Model., 1(3) (2018), 202-205.

The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method

Yıl 2019, Cilt: 2 Sayı: 2, 173 - 179, 20.12.2019

Ali Kurt Orkun Tasbozan Hulya Durur

https://doi.org/10.33401/fujma.562819

Cited By: 10

Öz

In this article, authors employed the new sub equation method to attain new traveling wave solutions of conformable time fractional partial differential equations. Conformable fractional derivative is a well behaved, applicable and understandable definition of arbitrary order derivation. Also this derivative obeys the basic properties that Newtonian concept satisfies. In this study authors obtained the exact solution for KDV equation where the fractional derivative is in conformable sense. New solutions are obtained in terms of the generalized version of the trigonometric functions.

Anahtar Kelimeler

Conformable Fractional Derivative, Exact solutions, The new sub-equation method

Kaynakça

[1] K. Oldham, J. Spanier, The Fractional Calculus, Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, 1974.
[2] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, 1993.
[3] I. Podlubny, Fractional Differential Equations, Academic Press,1999.
[4] A. A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
[5] A. Kurt, O. Tasbozan, Approximate analytical solution of the time fractional Whitham-Broer-Kaup equation using the homotopy analysis method, Int. J. Pure Appl. Math., 98(4) (2015), 503-510.
[6] O. Tasbozan, A. Esen, N. M. Yagmurlu, Y. Ucar, A numerical solution to fractional diffusion equation for force-free case, Abstr. Appl. Anal., 2013, Hindawi, (2013).
[7] C. Celik, M. Duman, Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. of Comput. Phys., 231(4) (2012), 1743-1750.
[8] Y. Cenesiz, A. Kurt, New fractional complex transform for conformable fractional partial differential equations, J. Appl. Math. Stat. Inf., 12(2) (2016), 41-47.
[9] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.
[10] M. Eslami, H. Rezazadeh, The first integral method for Wu-Zhang system with conformable time-fractional derivative, Calcolo, 53(3) (2016), 475-485.
[11] H. Aminikhah, A. R. Sheikhani, H. Rezazadeh, Sub-equation method for the fractional regularized long-wave equations with conformable fractional derivatives, Sci. Iran. Transaction B, Mech. Eng., 23(3) (2016), 1048.
[12] M. S. Osman, A. Korkmaz, H. Rezazadeh, M. Mirzazadeh, M. Eslami, Q. Zhou, The unified method for conformable time fractional Schrdinger equation with perturbation terms, Chinese J. Phys., 56(5) (2018), 2500-2506.
[13] Y. Cenesiz, D. Baleanu, A. Kurt, O. Tasbozan, New exact solutions of Burgers’ type equations with conformable derivative, Wave. Random. Complex, 27(1) (2017), 103-116.
[14] A. Kurt, O. Tasbozan, D. Baleanu, New solutions for conformable fractional Nizhnik-Novikov-Veselov system via G0=G expansion method and homotopy analysis methods, Opt. Quant. Electron., 49(10) (2017), 333.
[15] K. Hosseini, P. Mayeli, R. Ansari, Bright and singular soliton solutions of the conformable time-fractional Klein-Gordon equations with different nonlinearities, Wave. Random Complex , 28(3) (2018), 426-434.
[16] A. Korkmaz, K. Hosseini, Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods, Opt. Quant. Electron., 49(8) (2017), 278.
[17] H. Rezazadeh, H. Tariq, M. Eslami, M. Mirzazadeh, Q. Zhou, New exact solutions of nonlinear conformable time-fractional Phi-4 equation, Chinese J. Phys., 56(6) (2018), 2805-2816.
[18] H. Bulut, T.A. Sulaiman, H.M. Baskonus, H. Rezazadeh, M. Eslami, M. Mirzazadeh, Optical solitons and other solutions to the conformable space-time fractional Fokas-Lenells equation, Optik, 172 (2018), 20-27.
[19] H. Rezazadeh, S. M. Mirhosseini-Alizamini, M. Eslami, M. Rezazadeh, M. Mirzazadeh, S. Abbagari, New optical solitons of nonlinear conformable fractional Schr¨odinger-Hirota equation, Optik, 172 (2018), 545-553.
[20] I. E. Inan, Multiple soliton solutions of some nonlinear partial differential equations, Univers. J. Math. Appl., 1(4) (2018), 273-279.
[21] H. Rezazadeh, M. S. Osman, M. Eslami, M. Ekici, A. Sonmezoglu, M. Asma, W. A. M. Othman, B. R. Wong, M. Mirzazadeh, Q. Zhou, A. Biswas, M. Belic, Mitigating Internet bottleneck with fractional temporal evolution of optical solitons having quadratic-cubic nonlinearity, Optik, 164 (2018), 84-92.
[22] A. Biswas, M. O. Al-Amr, H. Rezazadeh, M. Mirzazadeh, M. Eslami, Q. Zhou, S. P. Moshokoa, M. Belic, Resonant optical solitons with dual-power law nonlinearity and fractional temporal evolution, Optik, 165 (2018), 233-239.
[23] H. Bulut, T. A. Sulaiman, H. M. Baskonus, Dark, bright optical and other solitons with conformable space-time fractional second-order spatiotemporal dispersion, Optik, 163 (2018), 1-7.
[24] M. H. Cherif, D. Ziane, Homotopy analysis Aboodh transform method for nonlinear system of partial differential Equations, Univers. J. Math. Appl., 1(4) (2018), 244-253.
[25] A. M. Wazwaz, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, App. Math. Comput., 184(2) (2007), 1002-1014.
[26] D. Ziane, T. M. Elzaki, M. Hamdi Cherif, Elzaki transform combined with variational iteration method for partial differential equations of fractional order, Fundam. J. Math. Appl., 1(1) (2018), 102-108.
[27] D. Feng, K. Li, On exact traveling wave solutions for (1+ 1) dimensional Kaup-Kupershmidt equation, Appl. Math., 2(6) (2011), 752-756.
[28] C. A. Gomez S, New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations, Appl. Math. Comput., 216(1) (2010), 241-250.
[29] F. Tascan, A. Akbulut, Construction of exact solutions to partial differential equations with CRE method, Commun. Adv. Math. Sci., 2(2) (2019), 105-113.
[30] M. H. Cherif, D. Ziane, Variational iteration method combined with new transform to solve fractional partial differential equations, Univers. J. Math. Appl., 1(2) (2018), 113-120.
[31] A. H. Salas, Solving the generalized Kaup-Kupershmidt equation, Adv. Studies Theor. Phys., 6(18) (2012), 879-885.
[32] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.
[33] H. Rezazadeh, A. Korkmaz, M. Eslami, J. Vahidi, R. Asghari, Traveling wave solution of conformable fractional generalized reaction Duffing model by generalized projective Riccati equation method, Opt. Quant. Electron., 50(3) (2018), 150.
[34] R. Polat, Finite difference solution to the space-time fractional partial differential-difference Toda lattice equation, J. Math. Sci. Model., 1(3) (2018), 202-205.

Toplam 34 adet kaynakça vardır.

Ayrıntılar

Birincil Dil	İngilizce
Konular	Matematik
Bölüm	Makaleler
Yazarlar	Ali Kurt 0000-0002-0617-6037 Orkun Tasbozan 0000-0001-5003-6341 Hulya Durur 0000-0002-9297-6873
Yayımlanma Tarihi	20 Aralık 2019
Gönderilme Tarihi	10 Mayıs 2019
Kabul Tarihi	26 Ekim 2019
Yayımlandığı Sayı	Yıl 2019 Cilt: 2 Sayı: 2

Kaynak Göster

APA	Kurt, A., Tasbozan, O., & Durur, H. (2019). The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method. Fundamental Journal of Mathematics and Applications, 2(2), 173-179. https://doi.org/10.33401/fujma.562819
AMA	Kurt A, Tasbozan O, Durur H. The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method. FUJMA. Aralık 2019;2(2):173-179. doi:10.33401/fujma.562819
Chicago	Kurt, Ali, Orkun Tasbozan, ve Hulya Durur. “The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method”. Fundamental Journal of Mathematics and Applications 2, sy. 2 (Aralık 2019): 173-79. https://doi.org/10.33401/fujma.562819.
EndNote	Kurt A, Tasbozan O, Durur H (01 Aralık 2019) The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method. Fundamental Journal of Mathematics and Applications 2 2 173–179.
IEEE	A. Kurt, O. Tasbozan, ve H. Durur, “The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method”, FUJMA, c. 2, sy. 2, ss. 173–179, 2019, doi: 10.33401/fujma.562819.
ISNAD	Kurt, Ali vd. “The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method”. Fundamental Journal of Mathematics and Applications 2/2 (Aralık 2019), 173-179. https://doi.org/10.33401/fujma.562819.
JAMA	Kurt A, Tasbozan O, Durur H. The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method. FUJMA. 2019;2:173–179.
MLA	Kurt, Ali vd. “The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method”. Fundamental Journal of Mathematics and Applications, c. 2, sy. 2, 2019, ss. 173-9, doi:10.33401/fujma.562819.
Vancouver	Kurt A, Tasbozan O, Durur H. The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method. FUJMA. 2019;2(2):173-9.