Microsoft Word - Ahmad _59-69_.docx R C O S T N Publisher: Research Council of Science and Technology, Biratnagar, Nepal p.59 BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (print), 2382-5340 (online) Journal homepage: http://nepjol.info/index.php/BIBECHANA On some nonlinear fractional PDEs in physics Jamshad Ahmad * , Syed Tauseef Mohyud-Din Department of Mathematics, Faculty of Sciences, HITEC University Taxila, Pakistan * Email: jamshadahmadm@gmail.com Accepted for publication: December 12, 2014 Abstract In this paper, we applied relatively new fractional complex transform (FCT) to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (PDEs) and Variational Iteration Method (VIM) is to find approximate solution of time- fractional Fornberg-Whitham and time-fractional Wu-Zhang equations. The results so obtained are re-stated by making use of inverse transformation which yields it in terms of original variables. It is observed that the proposed algorithm is highly efficient and appropriate for fractional PDEs arising in mathematical physics and hence can be extended to other problems of diversified nonlinear nature. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the proposed algorithm. DOI: http://dx.doi.org/10.3126/bibechana.v12i0.11687 © 2014 RCOST: All rights reserved. Keywords: Jumarie’s fractional derivatve; Fractional Fornberg-Whitham equation; fractional Wu- Zhang equations; variational iteration method; fractional complex transform. . 1. Introduction The nonlinear partial differential equations (NPDEs) are encountered in various disciplines, such as physics, mechanics, chemistry, biology, mathematics and engineering. Nonlinear partial differential equations [1-29] are of extreme importance. Recently, scientists have observed that number of real time problems is modeled by fractional nonlinear differential equations [4-5,7,10,19,21-25] which are very hard to tackle. Transform is an important method to solve mathematical problems. Recently the fractional complex transform [20-23] was suggested to convert fractional order differential equations with modified Riemann-Liouville derivatives [24-25] into integer order differential equations, and the resultant equations can be solved by different methods. This paper is devoted to the study of time-fractional Fornberg-Whitham equation, modified time fractional Fornberg-Whitham equation [6-12], time-fractional Wu-Zhang equation [24, 25]. The Fornberg–Whitham equation was first proposed for studying the qualitative behavior of wave breaking. The time fractional Fornberg–Whitham equation can be written as ���� − ���� + �� = ����� − ��� + 3����� , � > 0 , 0 < � ≤ 1, (1) Subject to the initial conditions Ahmad and Mohyud-Din/ BIBECHANA 12 (2015) 59-69: p. 60 ���, 0� = ����, and modifying the nonlinear term ��� in Eq. (1) by ����, He et al. proposed in [12] the modified time fractional Fornberg-Whitham equation ���� − ���� + �� = ����� − ���� + 3����� , � > 0 , 0 < � ≤ 1, (2) and time-fractional Wu-Zhang equation ���� + ��� + ��� + �� = 0 , ����+��� + ��� + �� = 0, 0 < � ≤ 1 (3) ���� + ����� + ����� + �� ����� + ���� + ���� + ����� = 0 with initial conditions ���, �, 0� = ����, ��, ���, �, 0� = ����, ��, ���, �, 0� = ����, ��. where w is the elevation of the water, u is the surface velocity of water along x -direction, and v is the surface velocity of water along y-direction. Wu and Zhang derived three sets of model equations for modeling nonlinear and dispersive long gravity waves travelling in two horizontal directions on shallow waters of uniform depth. Eq. (3) is one of these equations, Wu-Zhang equation (which describes (2+1)-dimensional dispersive long wave). The fractional derivatives are considered in the Jumarie sense. The basic motivation of this paper is the extension of a very reliable and efficient technique namely Variational Iteration Method using Complex Transform (VIMCT) to find approximate solutions of time-fractional Fornberg-Whitham and system of time-fractional Wu-Zhang equations. The convergence of the proposed variational iteration method using fractional derivative is addressed in [28-29]. It is observed that the proposed algorithms is fully synchronized with the complexity of fractional differential equations, Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the proposed algorithm. 2. Definitions Definition 2.1 Jumarie’s fractional derivative [24-25] is a modified Riemann-Liouville derivative defied as ������� = &'( ') � Γ�*α� + �x − t�*α*��f�t� − f�0��dt, α < 0 /0 , � Γ��*α� 11/ + �x − t�*α�f�t� − f�0��dt, 0 < α < 1,/0�f α*2�x��2 n ≤ α < 4 + 1, 4 ≥ 1. 6 (4) where �: 8 → 8, � → ���� denotes a continous (but not necessarily differentiable) function. Some useful formulas and results of Jumarie’s Modified Riemann-Liouville Derivatives are as follows: ���: = 0 , � > 0, : = :;4<�=4�. (5) ���>: ����? = : �������, � > 0, : = :;<4�=4�. (6) Ahmad and Mohyud-Din/ BIBECHANA 12 (2015) 59-69: p. 61 ��� �@ = Γ��A���BCD Γ��A@*�� , E > � > 0. (7) ���>����F���? = >�������? F��� + ���� >���F���?. (8) ���� ������ = ��̀���. �����. (9) 3. Variational Iteration Method (VIM) using Complex Transform The nonlinear differential equations [13-16] can be expressed in the operator form as �������, ��� + 8����, ��� + H����, ��� = 0. (10) where ��� is the time-fractional Jumarie’s fractional derivative , H���is the nonlinear operator and 8��� is some linear operator. The complex transform requires I = �� Using the basic properties of the fractional derivative [20], we can convert the fractional derivative into classical derivative. JDKJ�D = JKJL JLD J�D = σ JKJL, (11) where σ, is defined [20], Eq. (10) becomes σ �M�u��, ��� + 8�u��, ��� + H�u��, ��� = 0, (12) where �M = JKJM NM����, ���� + 8����, ���� + H����, ����=0, (13) where NM is the linear differential operator. According to Variational Iteration Method, we construct a correction functional as follows: �OA���, �� = �O��, �� + + P��, Q�>NR�O�x, Q��0 + 8�SO�x, Q� + H�SO��, ��?TQ, (14) where P is the general Lagrangian multiplier which can be indentified optimally by the variational theory ,the subscript 4 denotes the nth order approximation, and �SO is considered as a restricted variation, i.e. U�SO = 0. Its stationary conditions can be obtained as follows: P/�x, Q� = 0, 1 + P�x, Q� = 0. The Lagrange multiplier, therefore, can be obtained as P = −1, and the following variational iteration formula can be obtained as �OA���, �� = �O��, �� − + NR�O��, Q��0 + 8��O� + H��O�TQ. (15) Considering �0��, 0� = ���, 0�, we can easily find the components of the iterative formula defined in (15). Applying backward substitution to the computed components I = �� , we get Ahmad and Mohyud-Din/ BIBECHANA 12 (2015) 59-69: p. 62 ���, �� = limO→∞ �O��, ��. (16) 4. Numerical Applications Example 4.1 Consider the following time- fractional Fornberg-Whitham equation defined in Eq. (1) ���� − ���� + �� = ����� − ��� + 3����� , � > 0 , 0 < � ≤ 1, (17) Subject to the initial conditions ���, 0� = Z[\. Applying procedure defined in (11-15), u0 = e\̂, u��x, S� = − ��σ e\̂�−2 + S�, u��x, S� = �aσ e\̂�8 − 5S + S��, u��x, S� = − �deσ e\̂�−96 + 63S − 18S� + 2S��, ⋮. applying backward substitution u��x, t� = − ��σ e\̂�−2 + tα�, u��x, t� = �aσ e\̂�8 − 5tα + t�α�, U��x, t� = − �deσ e\̂�−96 + 63tα − 18t�α + 2t�α�, (18) ⋮. and so on. The exact solution of the time- fractional Fornberg-Whitham equation is obtained [6]. u�x, t� = e�\̂*\jk �. (19) Graphical representation of exact solution (19) and the approximate solutions (18) for ∝= 0.3,0.6,1. (a) α = 0.3 (b) α = 0.6 Ahmad and Mohyud-Din/ BIBECHANA 12 (2015) 59-69: p. 63 (c) α = 1 (d) Exact Solution Example 4.2 Consider the following time-fractional modified Fornberg-Whitham equation defined in Eq. (2) ���� − ���� + �� = ����� − ���� + 3����� , � > 0 , 0 < � ≤ 1, (20) Subject to the initial conditions ���, 0� = = sech��:��, where = = �q �√15 − 5�, : = ��0 �s10�5 − √15��. Applying procedure defined in (11-15) �0 = = sech��:��, u��x, S� = aσ coshw�cx� >coshx�cx�2c sinh�cx� coshq�cx�S + 2a�c sinh�cx� S32c�a sinh�cx� cosh��cx� + 60ac� sinh�cx� S, ⋮, and so on. Applying backward substitution u0 = a sech 2�cx�, u1�x, t� = a σcosh 7�cx� >cosh 5�cx� 2c sinh�cx� cosh 4�cx�tα + 2a2c sinh�cx� tα − 32c3a sinh�cx� cosh 2�cx� + 60ac3 sinh�cx� tα, (21) ⋮, and so on. The exact solution of the time- fractional modified Fornberg-Whitham equation is obtained [6]. u�x, t� = asech 2�c yx − �5 − z15�t�{ . (22) Graphical representation of exact solution (22) and the approximate solutions (21) for ∝= 0.3,0.6,1. Ahmad and Mohyud-Din/ BIBECHANA 12 (2015) 59-69: p. 64 (a) � = 0.3 (b) � = 0.6 (c) � = 1 (d) Exact Solution Example 4.3 Consider time- fractional Wu-Zhang equation, Dt αu + uux + vuy + wx = 0, Dt αv+uvx + vvy + wy = 0, (23) Dt αw + �uw�x + �vw�y + 1 3 �uxxx + uxyy + vxxy + vyyy� = 0, with initial conditions u�x, y, 0� = − k3Ak2b0 k1 + 2√3 3 k1 tanh�k1x + k2y�, v�x, y, 0� = b0 + 2√3 3 k2 tanh�k1x + k2y�, w�x, y, 0� = 2 3 �k1 2 + k2 2� sech 2�k1x + k2y�, where |0, }�, }� and }� are arbitrary constants. Applying the procedure defined above (10-15), we get u0�x, y, S� = − ~kA~\��~� + �√�� k� tanh�k�x + k�y�, v0�x, y, S� = b0 + �√�� k� tanh�k�x + k�y�, u0�x, y, S� = �� �k�� + k��� sech��k�x + k�y�, Ahmad and Mohyud-Din/ BIBECHANA 12 (2015) 59-69: p. 65 u1�x, y, S� = 1 3 σcosh 2�k1xAk2y� �−3k3 cosh 2�k1x + k2y�2 − 3k2 c cosh 2�k1x + k2y�2 + 2√3k1 2 sinh�k1x + k2y� cosh�k1x + k2y� + 2√3k1 2 Sk3�, v1�x, y, S� = 1 3σ cosh 2�k1xAk2y� �3c cosh 2�k1x + k2y�2 + 2√3 k2sinh�k1x + k2y� cosh�k1x + k2y� + 2√3k2Sk3�, w1�x, y, S� = − 2 3σ cosh 3�k1xAk2y� �2 sinh�k1x + k2y� Sk3k1 2 + 2 sinh�k1x + k2y� Sk3k2 2 − k1 2 cosh�k1x + k2y��, ⋮, and so on. Applying backward transformation u1�x, y, t� = 1 3σ cosh 2�k1xAk2y� �−3k3 cosh 2�k1x + k2y�2 − 3k2 c cosh 2�k1x + k2y�2 + 2√3k1 2 sinh�k1x + k2y� cosh�k1x + k2y� + 2√3k1 2 tαk3�, (24) v1�x, y, t� = 1 3σ cosh 2�k1xAk2y� �3c cosh 2�k1x + k2y�2 + 2√3 k2sinh�k1x + k2y� cosh�k1x + k2y� + 2√3k2tαk3�, (25) w��x, y, t� = − ��� ����k�~�/A~\�� �2 sinh�k�x + k�y� t�k�k�� + 2 sinh�k�x + k�y� t�k�k�� − k�� cosh�k�x + k�y��, (26) ⋮, and so on. Finally, we have u�x, y, t� = limn→∞ un�x, y, t�, v�x, y, t� = limn→∞ vn�x, y, t�, w�x, y, t� = limn→∞ wn�x, y, t�. The exact solution of time-fractional Wu-Zhang Equations [26], is given by u�x, y, t� = − k3Ak2b0 k1 + 2√3 3 k1 tanh�k1x + k2y + k3t�. (27) v�x, y, t� = b0 + 2√3 3 k2 tanh�k1x + k2y + k3t�. (28) w�x, y, t� = 2 3 �k1 2 + k2 2� sech 2�k1x + k2y + k3t�. (29) Graphical representation of exact solution (27) and the approximate solution (24) for ∝= 0.3,0.6,1. Ahmad and Mohyud-Din/ BIBECHANA 12 (2015) 59-69: p. 66 (a) α = 0.3 (b) α = 0.6 (c) α = 1 (d) Exact Solution Graphical representation of the exact solution (28) and the approximate solution (25) for ∝= 0.3,0.6,1. (a) α = 0.3 (b) α = 0.6 (c) α = 1 (d) Exact Solution Ahmad and Mohyud-Din/ BIBECHANA 12 (2015) 59-69: p. 67 Graphical representation of exact solution (29) and the approximate solution (26) for ∝= 0.3,0.6,1. (a) α = 0.3 (b) α = 0.6 (c) α = 1 (d) Exact Solution 5. Conclusions Applied fractional complex transform (FCT) proved very effective to convert the given partial differential equations (PDEs) into corresponding partial differential equations (PDEs) and the same is true for its subsequent effect in Variational Iteration Method (VIM) which was implemented on the transformed PDEs. 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