An Analytical Solution to Conformable Fractional Fokker-Planck Equation

Muammer Ayata; Ozan Özkan

doi:10.7212/karaelmasfen.941684

Research Article

Conformable Anlamında Kesirli Mertebeden Fokker-Planck Denkleminin Analitik Çözümü

Year 2022, Volume: 12 Issue: 1, 9 - 14, 01.06.2022

Muammer Ayata Ozan Özkan

Abstract

Bu çalışmada, istatistiksel fizikte önemli bir role sahip olan doğrusal olmayan kesirli Fokker Planck (FP) denklemi, yeni tanımlanmış
olan conformable Laplace ayrıştırma metodu (CLAM) ile ilk kez çözülmektedir. Bu yeni algoritma, conformable Laplace dönüşümü
ile Adomian ayrıştırma yöntemini birleştirmektedir. Çalışmamızda ilk olarak, kesirli türevin bazı temel tanımları ve teoremleri
conformable anlamında verilmiştir. Ardından CLAM’ ın genel algoritması anlatılmıştır. Bundan sonra, kullanılan metot grafikler
yardımıyla sayısal örnekle verilerek desteklenmiştir. Sayısal örnekten görüldüğü üzere, conformable Laplace ayrıştırma yöntemi güçlü,
güvenilir, kullanımı kolay ve kesirli mertebeden çok çeşitli kısmi türevli diferensiyel denklemlere uygulanabilir özelliklere sahiptir

Keywords

conformable fractional derivative, Laplace decomposition method, Fokker Planck equation

References

Abdeljawad, T. 2015. On Conformable Fractional Calculus. J. Comput. Applied Mathematics. 279: 57-66, DOI: 10.1016/j.cam.2014.10.016
Adomian, G. 1990. A Review of the Decomposition Method and Some Recent Results for Nonlinear Equations. Math. Comp. Modelling, 13(7): 17-43, DOI: 10.1016/0895-7177(90)90125-7
Atangana, A. 2015. Derivative with a New Parameter: Theory, Methods and Applications. Academic Press. DOI: https://doi.org/10.1016/c2014-0-04844-7
Atangana, A., Baleanu, D., Alsaedi, A. 2015. New Properties of Conformable Derivative. Open Math., DOI: 10.1515/math-2015-0081
Awan, MU., Noor, MA., Mihai, MV., Noor, KI. 2017. Conformable Fractional Hermite-Hadamard Inequalities via Preinvex Functions. Tbilisi Math. J., 10(4): 129-141, DOI: 10.1515/tmj-2017-0051
Ayata, M., Özkan, O. 2020. A New Application of Conformable Laplace Decomposition Method for Fractional Newell-Whitehead-Segel Equation. AIMS Math., 5(6): 7402-7412, DOI: 10.3934/math.2020474
Cherruault, Y., Saccomandi, G., Some, B. 1992. New Results for Convergence of Adomian's Method Applied to Integral Equations. Math. Comp. Mod., 16(2): 85-93, DOI: 10.1016/0895-7177(92)90009-A
Chung, WS. 2015. Fractional Newton Mechanics with Conformable Fractional Derivative. J. Comp. Appl. Math., 290, 150-158, DOI: 10.1016/j.cam.2015.04.049
Eslami, M., Rezazadeh, H. 2016. The First Integral Method for Wu–Zhang System with Conformable Time-Fractional Derivative. Calcolo, 53(3), 475-485, DOI: 10.1007/s10092-015-0158-8
Eslami M, Taleghani SA, 2019. Differential Transform Method for Conformable Fractional Partial Differential Equations. Iranian J. Numerical Analy. Opt., 9(2), 17-29, DOI: 10.22067/IJNAO.V9I2.67976
Frank, TD. 2004. Stochastic Feedback, Nonlinear Families of Markov Processes, and Nonlinear Fokker–Planck Equations. Physica A: Statistical Mech. its Appl., 331(3-4), 391-408, DOI: 10.1016/j.physa.2003.09.056
Geng, F., Cui, M. 2011. A Novel Method for Nonlinear Two-Point Boundary Value Problems: Combination of ADM and RKM. Appl. Math. Comp., 217(9), 4676-4681, DOI: 10.1016/j.amc.2010.11.020
Hammad, MA., Khalil, R. 2014. Abel's Formula and Wronskian for Conformable Fractional Differential Equations. Inter. J. Diff. Eqns. Appl., 13(3), DOI: 10.12732/ijdea.v13i3.1753
Hosseini, SG., Abbasbandy, S. 2015. Solution of Lane-Emden Type Equations By Combination Of The Spectral Method and Adomian Decomposition Method. Math. Probl. Eng., 2015, DOI: 10.1155/2015/534754
Jordan, R., Kinderlehrer, D., Otto, F. 1998. The Variational Formulation of the Fokker-Planck Equation. SIAM J. Math. Anal., 29(1), 1-17, DOI: 10.1137/S0036141096303359
Keyanpour, M., Mahmoudi, A. 2012. A Hybrid Method for Solving Optimal Control Problems. IAENG Int. J. Appl. Math., 42(2), 80-86
Khalil, R., Al Horani, M., Yousef, A., Sababheh, M. 2014. A New Definition of Fractional Derivative. J. Comput. Appl. Math., 264, 65-70, DOI: 10.1016/j.cam.2014.01.002
Kurt, A., Tasbozan, O., Baleanu, D. 2017. New Solutions for Conformable Fractional Nizhnik–Novikov–Veselov System via G'/G Expansion Method and Homotopy Analysis Methods. Optic. Quantum Elect., 49(10), 333, DOI: doi.org/10.1007/s11082-017-1163-8
Ladrani, FZ., Cherif, AB. 2020. Oscillation Tests for Conformable Fractional Differential Equations with Damping. Punjab Univ. J. Math., 52(2), 73-82
Miller, KS., Ross, B. 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley
Momani, S., Odibat, Z. 2007. Homotopy Perturbation Method for Nonlinear Partial Differential Equations of Fractional Order. Physics Letters A, 365(5-6), 345-350, DOI: 10.1016/j.physleta.2007.01.046
Nourazar, S., Ramezanpour, M., Doosthoseini, A. 2013. A New Algorithm to Solve the Gas Dynamics Equation: An application of the Fourier Transform Adomian Decomposition Method. Appl. Math. Sci., 7(86), 4281-4286, DOI: 10.12988/ams.2013.33147
Özkan, O., Kurt, A. 2018a. The Analytical Solutions for Conformable Integral Equations and Integro-Differential Equations by Conformable Laplace Transform. Opt. Quantum Elect., 50(2), 81, DOI: 10.1007/s11082-018-1342-2
Özkan, O., Kurt, A. 2018b. On Conformable Double Laplace Transform. Opt. Quantum Elect., 50(2), 103. DOI: 10.1007/s11082-018-1372-9
Özkan, O., Kurt, A. 2019. Exact Solutions of Fractional Partial Differential Equation Systems with Conformable Derivative. Filomat, 33(5), 1313-1322, DOI: 10.2298/FIL1905313O
Podlubny, I. 1999. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Elseiver, DOI: 10.1016/s0076-5392(99)x8001-5
Rach, R. 1984. A Convenient Computational Form for the Adomian Polynomials. J. Math. Analy. Appl., 102(2), 415-419, DOI: 10.1016/0022-247X(84)90181-1
Ray, SS., Bera, RK. 2005. An Approximate Solution of a Nonlinear Fractional Differential Equation by Adomian Decomposition Method. Appl. Math. Comput., 167(1), 561-571, DOI: 10.1016/j.amc.2004.07.020
Ray, SS. 2016. New Analytical Exact Solutions of Time Fractional Kdv–Kzk Equation By Kudryashov Methods. Chinese Physics B, 25(4), 040204, DOI: 10.1088/1674-1056/25/4/040204
Risken, H., Frank, T. 1989. Methods of Solution and Applications. Springer Series Synergetics, 18
Qi, Y., Wang, X. 2020. Asymptotical Stability Analysis of Conformable Fractional Systems. J. Taibah Univ. Sci., 14(1), 44-49, DOI: 10.1080/16583655.2019.1701390
Tayyan, BA., Sakka, AH. 2020. Lie Symmetry Analysis of Some Conformable Fractional Partial Differential Equations. Arabian J. Math., 9(1), 201-212, DOI: 10.1007/s40065-018-0230-8
Wu, GC., Lee, EWM. 2010. Fractional Variational Iteration Method and its Application. Physics Letters A, 374(25): 2506-2509, DOI: 10.1016/j.physleta.2010.04.034
Zafar, A., Rezazadeh, H., Bekir, A., Malik, A. 2019. Exact Solutions of 3-Dimensional Fractional mKdV Equations in Conformable Form via exp(-ϕ(τ)) Expansion Method. SN Appl. Sci., 1(11), 1436, DOI: 10.1007/s42452-019-1424-1
Zhang, S., Zhang, HQ. 2011. Fractional Sub-Equation Method and its Applications to Nonlinear Fractional PDEs. Physics Letters A, 375(7), 1069-1073, DOI: 10.1016/j.physleta.2011.01.029
Zhao, D., Luo, M. 2017. General Conformable Fractional Derivative and its Physical Interpretation. Calcolo, 54(3), 903-917, DOI: 10.1007/s10092-017-0213-8

An Analytical Solution to Conformable Fractional Fokker-Planck Equation

Year 2022, Volume: 12 Issue: 1, 9 - 14, 01.06.2022

Muammer Ayata Ozan Özkan

Abstract

It is the first time in this work that a newly defined conformable Laplace decomposition method (CLDM) is applied to nonlinear
fractional Fokker Planck (FP) equation which has a major role in statistical physics. This new algorithm combines conformable
Laplace transform and Adomian decomposition method. First, some basic theorems and definitions of fractional derivative are given
in conformable sense. Then, the general algorithm of CLDM is presented. After that, the presented method is supported by numerical
example by the aid of figures. As is seen from the numerical example, conformable Laplace decomposition method is strong, easy to
use, reliable and applicable to a wide variety of fractional PDEs.

Keywords

conformable fractional derivative, Laplace decomposition method, Fokker Planck equation

References

Abdeljawad, T. 2015. On Conformable Fractional Calculus. J. Comput. Applied Mathematics. 279: 57-66, DOI: 10.1016/j.cam.2014.10.016
Adomian, G. 1990. A Review of the Decomposition Method and Some Recent Results for Nonlinear Equations. Math. Comp. Modelling, 13(7): 17-43, DOI: 10.1016/0895-7177(90)90125-7
Atangana, A. 2015. Derivative with a New Parameter: Theory, Methods and Applications. Academic Press. DOI: https://doi.org/10.1016/c2014-0-04844-7
Atangana, A., Baleanu, D., Alsaedi, A. 2015. New Properties of Conformable Derivative. Open Math., DOI: 10.1515/math-2015-0081
Awan, MU., Noor, MA., Mihai, MV., Noor, KI. 2017. Conformable Fractional Hermite-Hadamard Inequalities via Preinvex Functions. Tbilisi Math. J., 10(4): 129-141, DOI: 10.1515/tmj-2017-0051
Ayata, M., Özkan, O. 2020. A New Application of Conformable Laplace Decomposition Method for Fractional Newell-Whitehead-Segel Equation. AIMS Math., 5(6): 7402-7412, DOI: 10.3934/math.2020474
Cherruault, Y., Saccomandi, G., Some, B. 1992. New Results for Convergence of Adomian's Method Applied to Integral Equations. Math. Comp. Mod., 16(2): 85-93, DOI: 10.1016/0895-7177(92)90009-A
Chung, WS. 2015. Fractional Newton Mechanics with Conformable Fractional Derivative. J. Comp. Appl. Math., 290, 150-158, DOI: 10.1016/j.cam.2015.04.049
Eslami, M., Rezazadeh, H. 2016. The First Integral Method for Wu–Zhang System with Conformable Time-Fractional Derivative. Calcolo, 53(3), 475-485, DOI: 10.1007/s10092-015-0158-8
Eslami M, Taleghani SA, 2019. Differential Transform Method for Conformable Fractional Partial Differential Equations. Iranian J. Numerical Analy. Opt., 9(2), 17-29, DOI: 10.22067/IJNAO.V9I2.67976
Frank, TD. 2004. Stochastic Feedback, Nonlinear Families of Markov Processes, and Nonlinear Fokker–Planck Equations. Physica A: Statistical Mech. its Appl., 331(3-4), 391-408, DOI: 10.1016/j.physa.2003.09.056
Geng, F., Cui, M. 2011. A Novel Method for Nonlinear Two-Point Boundary Value Problems: Combination of ADM and RKM. Appl. Math. Comp., 217(9), 4676-4681, DOI: 10.1016/j.amc.2010.11.020
Hammad, MA., Khalil, R. 2014. Abel's Formula and Wronskian for Conformable Fractional Differential Equations. Inter. J. Diff. Eqns. Appl., 13(3), DOI: 10.12732/ijdea.v13i3.1753
Hosseini, SG., Abbasbandy, S. 2015. Solution of Lane-Emden Type Equations By Combination Of The Spectral Method and Adomian Decomposition Method. Math. Probl. Eng., 2015, DOI: 10.1155/2015/534754
Jordan, R., Kinderlehrer, D., Otto, F. 1998. The Variational Formulation of the Fokker-Planck Equation. SIAM J. Math. Anal., 29(1), 1-17, DOI: 10.1137/S0036141096303359
Keyanpour, M., Mahmoudi, A. 2012. A Hybrid Method for Solving Optimal Control Problems. IAENG Int. J. Appl. Math., 42(2), 80-86
Khalil, R., Al Horani, M., Yousef, A., Sababheh, M. 2014. A New Definition of Fractional Derivative. J. Comput. Appl. Math., 264, 65-70, DOI: 10.1016/j.cam.2014.01.002
Kurt, A., Tasbozan, O., Baleanu, D. 2017. New Solutions for Conformable Fractional Nizhnik–Novikov–Veselov System via G'/G Expansion Method and Homotopy Analysis Methods. Optic. Quantum Elect., 49(10), 333, DOI: doi.org/10.1007/s11082-017-1163-8
Ladrani, FZ., Cherif, AB. 2020. Oscillation Tests for Conformable Fractional Differential Equations with Damping. Punjab Univ. J. Math., 52(2), 73-82
Miller, KS., Ross, B. 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley
Momani, S., Odibat, Z. 2007. Homotopy Perturbation Method for Nonlinear Partial Differential Equations of Fractional Order. Physics Letters A, 365(5-6), 345-350, DOI: 10.1016/j.physleta.2007.01.046
Nourazar, S., Ramezanpour, M., Doosthoseini, A. 2013. A New Algorithm to Solve the Gas Dynamics Equation: An application of the Fourier Transform Adomian Decomposition Method. Appl. Math. Sci., 7(86), 4281-4286, DOI: 10.12988/ams.2013.33147
Özkan, O., Kurt, A. 2018a. The Analytical Solutions for Conformable Integral Equations and Integro-Differential Equations by Conformable Laplace Transform. Opt. Quantum Elect., 50(2), 81, DOI: 10.1007/s11082-018-1342-2
Özkan, O., Kurt, A. 2018b. On Conformable Double Laplace Transform. Opt. Quantum Elect., 50(2), 103. DOI: 10.1007/s11082-018-1372-9
Özkan, O., Kurt, A. 2019. Exact Solutions of Fractional Partial Differential Equation Systems with Conformable Derivative. Filomat, 33(5), 1313-1322, DOI: 10.2298/FIL1905313O
Podlubny, I. 1999. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Elseiver, DOI: 10.1016/s0076-5392(99)x8001-5
Rach, R. 1984. A Convenient Computational Form for the Adomian Polynomials. J. Math. Analy. Appl., 102(2), 415-419, DOI: 10.1016/0022-247X(84)90181-1
Ray, SS., Bera, RK. 2005. An Approximate Solution of a Nonlinear Fractional Differential Equation by Adomian Decomposition Method. Appl. Math. Comput., 167(1), 561-571, DOI: 10.1016/j.amc.2004.07.020
Ray, SS. 2016. New Analytical Exact Solutions of Time Fractional Kdv–Kzk Equation By Kudryashov Methods. Chinese Physics B, 25(4), 040204, DOI: 10.1088/1674-1056/25/4/040204
Risken, H., Frank, T. 1989. Methods of Solution and Applications. Springer Series Synergetics, 18
Qi, Y., Wang, X. 2020. Asymptotical Stability Analysis of Conformable Fractional Systems. J. Taibah Univ. Sci., 14(1), 44-49, DOI: 10.1080/16583655.2019.1701390
Tayyan, BA., Sakka, AH. 2020. Lie Symmetry Analysis of Some Conformable Fractional Partial Differential Equations. Arabian J. Math., 9(1), 201-212, DOI: 10.1007/s40065-018-0230-8
Wu, GC., Lee, EWM. 2010. Fractional Variational Iteration Method and its Application. Physics Letters A, 374(25): 2506-2509, DOI: 10.1016/j.physleta.2010.04.034
Zafar, A., Rezazadeh, H., Bekir, A., Malik, A. 2019. Exact Solutions of 3-Dimensional Fractional mKdV Equations in Conformable Form via exp(-ϕ(τ)) Expansion Method. SN Appl. Sci., 1(11), 1436, DOI: 10.1007/s42452-019-1424-1
Zhang, S., Zhang, HQ. 2011. Fractional Sub-Equation Method and its Applications to Nonlinear Fractional PDEs. Physics Letters A, 375(7), 1069-1073, DOI: 10.1016/j.physleta.2011.01.029
Zhao, D., Luo, M. 2017. General Conformable Fractional Derivative and its Physical Interpretation. Calcolo, 54(3), 903-917, DOI: 10.1007/s10092-017-0213-8

There are 36 citations in total.

Details

Primary Language	English
Subjects	Engineering
Journal Section	Research Articles
Authors	Muammer Ayata 0000-0001-9436-6414 Ozan Özkan 0000-0001-6430-1126
Publication Date	June 1, 2022
Published in Issue	Year 2022 Volume: 12 Issue: 1

Cite

APA	Ayata, M., & Özkan, O. (2022). An Analytical Solution to Conformable Fractional Fokker-Planck Equation. Karaelmas Fen Ve Mühendislik Dergisi, 12(1), 9-14. https://doi.org/10.7212/karaelmasfen.941684
AMA	Ayata M, Özkan O. An Analytical Solution to Conformable Fractional Fokker-Planck Equation. Karaelmas Fen ve Mühendislik Dergisi. June 2022;12(1):9-14. doi:10.7212/karaelmasfen.941684
Chicago	Ayata, Muammer, and Ozan Özkan. “An Analytical Solution to Conformable Fractional Fokker-Planck Equation”. Karaelmas Fen Ve Mühendislik Dergisi 12, no. 1 (June 2022): 9-14. https://doi.org/10.7212/karaelmasfen.941684.
EndNote	Ayata M, Özkan O (June 1, 2022) An Analytical Solution to Conformable Fractional Fokker-Planck Equation. Karaelmas Fen ve Mühendislik Dergisi 12 1 9–14.
IEEE	M. Ayata and O. Özkan, “An Analytical Solution to Conformable Fractional Fokker-Planck Equation”, Karaelmas Fen ve Mühendislik Dergisi, vol. 12, no. 1, pp. 9–14, 2022, doi: 10.7212/karaelmasfen.941684.
ISNAD	Ayata, Muammer - Özkan, Ozan. “An Analytical Solution to Conformable Fractional Fokker-Planck Equation”. Karaelmas Fen ve Mühendislik Dergisi 12/1 (June 2022), 9-14. https://doi.org/10.7212/karaelmasfen.941684.
JAMA	Ayata M, Özkan O. An Analytical Solution to Conformable Fractional Fokker-Planck Equation. Karaelmas Fen ve Mühendislik Dergisi. 2022;12:9–14.
MLA	Ayata, Muammer and Ozan Özkan. “An Analytical Solution to Conformable Fractional Fokker-Planck Equation”. Karaelmas Fen Ve Mühendislik Dergisi, vol. 12, no. 1, 2022, pp. 9-14, doi:10.7212/karaelmasfen.941684.
Vancouver	Ayata M, Özkan O. An Analytical Solution to Conformable Fractional Fokker-Planck Equation. Karaelmas Fen ve Mühendislik Dergisi. 2022;12(1):9-14.

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