Conformable Kısmi Diferansiyel Denklemlerin Homotopi Analiz Yöntemi ile Nümerik Çözümleri
Bu makalede, kesirli Wu-Zhang sisteminin ve birleştirilmiş KdV-mKdV denklemi denkleminin nümerikçözümlerini elde etmek için Homotopi Analiz Yöntemi (HAM) uygulandı. Elde edilen sonuçlar, analitikçözümlerler ile karşılaştırıldı.
Numerical Solutions of Conformable Partial Differential Equations By Homotopy Analysis Method
In this paper, the Homotopy Analysis Method (HAM) is applied to the fractional Wu-Zhang system and combined KdV-mKdV equation to obtain theirnumerical solutions. The results were compared with analytical solutions.
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- 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.