College of Engineering Mechanics and Soft Materials, Hohai University, Jiangning, Nanjing, China b

Transcription

1 THERMA SCIENCE: Year 28, Vol. 22, Suppl., pp. S65-S75 S65 A MODIFICATION FRACTIONA VARIATIONA ITERATION METHOD FOR SOVING NON-INEAR GAS DYNAMIC AND COUPED KdV EQUATIONS INVOVING OCA FRACTIONA OPERATORS b Dumitru BAEANU a,b,c, Haan Kamil JASSIM d, and Haib KHAN a,e * a College of Engineering Mechanic and Soft Material, Hohai Univerit, Jiangning, Nanjing, China b Department of Mathematic, Facult of Art and Science, Cankaa Univerit, Ankara, Turke c Intitute of Space Science, Magurele-Bucharet, Romania d Department of Mathematic, Facult of Education for Pure Science, Univerit of Thi-Qar, Nairiah, Iraq e Department of Mathematic, Shaheed BB Univerit, Sheringal, Dir Upper, Khber Pakhtunkhwa, Pakitan Original cientific paper Introduction In thi paper, we appl a new technique, namel local fractional variational iteration tranform method on homogeneou/non-homogeneou non-linear ga dnamic and coupled KdV equation to obtain the analtical approimate olution. The iteration procedure i baed on local fractional derivative and integral operator. Thi method i the combination of the local fractional aplace tranform and variational iteration method. The method in general i ea to implement and ield good reult. Illutrative eample are included to demontrate the validit and applicabilit of the new technique. Ke word: coupled KdV equation, non-linear ga dnamic equation, local fractional variational iteration method, local fractional aplace tranform, local fractional operator The variational iteration method wa firt propoed b He [, 2] and wa applied to deal with Helmholtz equation in [], Burger and coupled Burger equation in [4], Klein-Gordon equation in [5], in KdV in [6], the ocillation equation in [7], Schrodinger equation in [8], diffuion equation in [9], Bernoulli equation in [], and other. The etended variational iteration method, called the fractional variational iteration method, wa developed and applied to handle ome fractional differential equation within the modified Riemann-iouville derivative in [-5]. More recentl, the local fractional variational iteration method, initiated in [6], wa ued to find the non-differentiable olution for the heat conduction equation, Poion equation in [7], coupled KdV equation in [8], damped and diipative wave equation in [9], Fokker-Planck equation in [2], and non-linear PDE in [2] with local fractional derivative operator. * Correponding author, haibkhan@ahoo.com

2 S66 Baleanu, D., et al.: A Modification Fractional Variational Iteration Method for Solving... THERMA SCIENCE: Year 28, Vol. 22, Suppl., pp. S65-S75 In recent ear, a man of approimate and analtical method have been utilized to olve the PDE with local fractional derivative operator uch a the local fractional Adomian decompoition method in [22, 2], local fractional differential tranform method in [24-26], local fractional erie epanion method in [27], local fractional Sumudu tranform method in [28], local fractional aplace tranform method in [29], local fractional reduced differential tranform method in [, 2], local fractional aplace variational iteration method in []. The tandard form of non-linear ga dnamic equation involving local fractional derivative operator can be written: 2 (, (, u u 2 u (, u (, g(,, < ubject to the initial condition (, φ ( u (2 The KdV equation decribe the theor of water wave in hallow channel. It i a non-linear equation which ehibit pecial olution, known a Solution, which are table and do not dipere with time. The coupled KdV equation involving local fractional derivative operator can be written: u, u, u, u, 2 u (, 2 v (, v, v, v, v, 2 v (, 2 u (, ubject to the initial condition (, κ (, κ u v In thi paper, our aim are to preent the coupling method of local fractional aplace tranform and variational iteration method, which i called a the local fractional variational iteration tranform method, and to ue it to olve the non-linear ga dnamic and coupled KdV equation with local fractional derivative. Anali of the method We conider a general non-linear local PDE: 2 u, Ru, Nu, g,, < (5 n n where a /, n N i the linear FDO, R denote a lower order FDO, N repreented the general non-linear FDO, and g (, i the non-differentiable ource term. Appling the Yang-aplace tranform (denoted b on both ide of eq. (5, we get: { u (, } { Ru (, } { Nu (, } { (, } g (6 Uing the propert of the Yang-aplace tranform, we have: ( (4

3 THERMA SCIENCE: Year 28, Vol. 22, Suppl., pp. S65-S75 S67 or ( n ( n 2 ( { } { g(, Ru (, Nu (, } ( n n u, u, u, u, (7 u, u, u, u, { } ( [( n ] 2 n { g(, Ru (, Nu (, n } (8 Operating with the Yang-aplace invere on both ide of eq. (8 give: ( n u, u, u, u, ( Γ ( Γ ( n { g(, Ru (, Nu (, n } ( n (9 Deriving both ide eq. (9 with repect to, we have: { g(, Ru (, Nu (, n } u, ( n2 ( ( n u (, u (, Γ ( n 2 We now tructure the correctional local fractional function in the form: ( m (, m(, Γ ( { g(, R um(, N um(, k } u u um, λ( ( d ( n2 Γ ( [( n ] ( um (, ( um (, Γ ( n 2 Making the local fractional variation, we get: ( (, m(, Γ ( { g(, R um(, N um(, k } δ u δ u δ m um, λ( ( d n 2 ( (2 [( n ] ( um (, ( um (, Γ ( n 2 The etremum condition of u (, n i given: (

4 S68 Baleanu, D., et al.: A Modification Fractional Variational Iteration Method for Solving... THERMA SCIENCE: Year 28, Vol. 22, Suppl., pp. S65-S75 ( δ um, ( In view of eq. (, we have the following tationar condition: Thi i turn give: ( λ λ, ( λ Subtituting eq. (5 into eq. (, we obtained: ( m (, m(, Γ ( { g(, R um(, N um(, k } u u u ( n2 [( n ] ( um (, ( um (, Γ ( n 2 m, ( d (4 (5 (6 Application Finall, the olution u (, i given: (, lim (, u u (7 m Eample. Conider the following homogeneou non-linear ga dnamic equation involving local fractional derivative operator: and the initial condition 2 (, (, u u u (, u (, 2 m (, (8 u E (9 In view of eq. (6 and (8 the local fractional iteration algorithm can be written: un (, un(, 2 ( τ, ( un u n un( τ, un( τ, ( dτ τ τ 2 (2 We can ue the initial condition to elect u (, u(, E. Uing thi election into the correction functional (2 give the following ucceive approimation:

5 THERMA SCIENCE: Year 28, Vol. 22, Suppl., pp. S65-S75 S69 u E (2, u(, u(, 2 ( τ, ( u u u( τ, u( τ, ( dτ τ τ 2 E ( ( (22 u2(, u(, 2 ( τ, ( u u u( τ, u( τ, ( dτ τ τ 2 E ( n (, ( u E Finall, the olution u (, i given: and in cloed form: 2 ( ( 2 k (, lim n (, ( u u E n ( k (2 n k (24 k (25 k ( k (, u E E E (26 Eample 2. Conider the following non-homogeneou non-linear ga dnamic equation involving local fractional derivative operator: 2 (, (, u u 2 u u E with the initial condition: (, (, (, (27 u E (28 Appling eq. (6 and (2, we obtain the correction function: un (, un(, 2 ( τ, ( un un un( τ, un( τ, ( dτ τ τ 2 (29

6 S7 Baleanu, D., et al.: A Modification Fractional Variational Iteration Method for Solving... THERMA SCIENCE: Year 28, Vol. 22, Suppl., pp. S65-S75 We can ue the initial condition to elect u (, u(, E Uing thi election into the correction functional (29 give the following ucceive approimation: 2 ( τ, u, E ( u(, u(, ( u u u ( τ, u ( τ, τ 2 ( dτ E ( τ E ( E ( τ ( dτ E ( E E E 2 ( τ, ( u2(, u(, ( τ ( ( u u u(, u(, τ τ τ τ 2 ( dτ E Finall, the olution u (, i given: E (2 u, E ( n (, lim n (, u u E (4 n Eample. Conider the tem of local fractional coupled KdV equation with local fractional derivative: u, u, u, u, 2 u (, 2 v (, v, v, u, v, 2 v (, 2 u (, ubject to the initial condition: (, ( (, ( u E v E (5 (6

7 THERMA SCIENCE: Year 28, Vol. 22, Suppl., pp. S65-S75 S7 Appling local fractional aplace tranform on eq. (5 and uing the initial condition (6, we have: u, u, u, { u (, } E ( 2 u (, 2 v (, (7 v (, v (, v (, { v (, } E ( 2 v (, 2 u (, Operating with the local fractional aplace tranform invere on both ide of eq. (7 we obtain: u, u, u, u (, E ( 2 u (, 2 v (, v, v, v, v (, E ( 2 v (, 2 u (, Deriving both ide of eq. (8 with repect to, we get: u, u, u, u, 2 u (, 2 v (, (9 v (, v (, v (, v (, 2 v (, 2 u (, Making the correction function i given: um (, um(, ( τ ( ( τ 2 um, um, 2 u m (, τ um ( τ, τ τ um ( τ, 2 vm (, vm (, vm(, ( τ ( τ vm, vm, 2 v m (, τ vm ( τ, τ τ vm ( τ, 2 vm (, ( d τ ( dτ We can ue the initial condition to elect u(, E(, v(, E(. Uing thi election into the correction functional (4 give the following ucceive approimation: (8 (4

8 S72 Baleanu, D., et al.: A Modification Fractional Variational Iteration Method for Solving... THERMA SCIENCE: Year 28, Vol. 22, Suppl., pp. S65-S75 ( u E, ( v E, u(, u(, ( τ ( ( τ 2 u, u, 2 u (, τ u ( τ, τ τ u ( τ, 2 v (, v(, v(, ( τ ( ( τ v, v, 2 v (, τ v ( τ, τ τ v ( τ, 2 v (, 2 ( 2 E E E ( d τ ( dτ ( d τ Γ ( τ 2E ( 2 ( 2 E E E ( d τ Γ ( τ 2E E ( E ( ( d τ E ( E ( ( d τ E ( Γ ( E ( (

9 THERMA SCIENCE: Year 28, Vol. 22, Suppl., pp. S65-S75 S7 u2(, u(, ( τ ( ( τ 2 u, u, 2 u (, τ u ( τ, τ τ u ( τ, 2 v (, v2(, v(, ( τ ( ( τ v, v, 2 v (, τ v ( τ, τ τ v ( τ, 2 u (, E ( E ( τ E ( E ( E Γ ( τ Γ ( E ( E ( τ E ( E ( E τ Γ ( τ E E E Γ ( Γ( Γ ( τ E E E ( d τ ( dτ ( dτ ( 2 ( dτ ( dτ Γ ( Γ( Γ ( 2 E ( Γ ( Γ ( 2 2 E ( Γ ( Γ ( 2 m k um (, E ( Γ k (, ( v E m k k m k ( k ( dτ

10 S74 Baleanu, D., et al.: A Modification Fractional Variational Iteration Method for Solving... THERMA SCIENCE: Year 28, Vol. 22, Suppl., pp. S65-S75 Therefore, the erie olution can be written in the form: (, lim m (, ( m (, lim m (, ( u u E v v E m (4 Concluion In thi work, local fractional variational iteration tranform method ha been uccefull applied to finding the non-differentiable olution of non-linear ga dnamic and coupled KdV equation involving local fractional operator. The method i ver powerful and efficient in finding analtical a well a numerical olution for wide clae of linear and non-linear local fractional PDE. Acknowledgment Reference Thi work wa upported b the China Government Young Talent Program. [] He, J. H., Approimate Analtical Solution for Seepage Flow with Fractional Derivative in Porou Media, Computer Method in Applied Mechanic and Engineering, 67 (998-2, pp [2] He, J. H., Variational Iteration Method a Kind of Non-inear Analtical Technique: Some Eample, International Journal of Non-inear Mechanic, 4 (999, 4, pp [] Momani, S., et al., Application of He Variational Iteration Method to Helmholtz Equation, Chao, Soliton & Fractal, 27 (26, 5, pp. 9-2 [4] Abdou, M. A., et al., Variational Iteration Method for Solving Burger and Coupled Burger Equation, Journal of Computational and Applied Mathematic, 8 (25, 2, pp [5] Abba Band, S., et al., Numerical Solution of Non-inear Klein-Gordon Equation b Variational Iteration Method, International Journal for Numerical Method in Engineering, 7 (27, 7, pp [6] Mohud-Din, S. T., et al., Traveling Wave Solution of Seventh-Order Generalized KdV Equation Uing He Polnomial, International Journal of Nonlinear Science and Numerical Simulation, (29, 2, pp [7] Marinca, V., et al., Application of the Variational Iteration Method to Some Nonlinear One-Dimenional Ocillation, Meccanica, 4 (28,, pp [8] Wazwaz, A. M., A Stud on inear and Nonlinear Schroedinger Equation b the Variational Iteration Method, Chao, Soliton & Fractal, 7 (28, 4, pp [9] Da, S., et al., Approimate Solution of Fractional Diffuion Equation Reviited, International Review of Chemical Engineering, 4 (22 5, pp [] Hritov, J., An Eercie with the He Variation Iteration Method to a Fractional Bernoulli Equation Ariing in a Tranient Conduction with a Non-inear Boundar Heat Flu, International Review of Chemical Engineering, 4 (22, 5, pp [] Wu, G. C., et al., Fractional Variational Iteration Method and It Application, Phic etter A: General, Atomic and Solid State Phic, 74 (2, 25, pp [2] He, J. H., A Short Remark on Fractional Variational Iteration Method, Phic etter A: General, Atomic and Solid State Phic, 75 (2, 8, pp [] Baleanu, D., et al., Improved (G /G-Epanion Method for the Time-Fractional Biological Population Model and Cahn-Hilliard Equation, Journal of Computational and Nonlinear Dnamic, (25, 5, 56 [4] Mutafa, I., et al., An Approimate Solution of Fractional Cable Equation b HAM, Boundar Value Problem, 24 (24, 58, pp. - [5] Bulent, K., et al., The Firt Integral Method for the Time Fractional Kaup-Bouineq Stem with Time Dependent Coefficient, Applied Mathematic and Computation, 254 (25, C, pp [6] Yang, X. J., et al., Fractal Heat Conduction Problem Solved b ocal Fractional Variation Iteration Method, Thermal Science, 7 (2, 2, pp

11 THERMA SCIENCE: Year 28, Vol. 22, Suppl., pp. S65-S75 S75 [7] Chen,., et al., ocal Fractional Variational Iteration Method for ocal Fractional Poion Equation in Two Independent Variable, Abtract and Applied Anali, 24 (24, ID4842 [8] Jafari, H., et al., Approimate Solution for Nonlinear Ga Dnamic and Coupled KdV Equation Involving ocal Fractional Operator, Journal of Zanko Sulaimani Part A, 8 (26,, pp [9] Su, W. H., et al., Damped Wave Equation and Diipative Wave Equation in Fractal String within the ocal Fractional Variational Iteration Method, Fied Point Theor and Application, 89 (2,, pp. - [2] Jaim, H. K., New Approache for Solving Fokker Planck Equation on Cantor Set within ocal Fractional Operator, Journal of Mathematic, 25 (25, ID [2] Jafari, H., et al., ocal Fractional Variational Iteration Method for Nonlinear Partial Differential Equation within ocal Fractional Operator, Application and Applied Mathematic, (25, 2, pp [22] Jafari, H., Jaim, H. K., ocal Fractional Adomian Decompoition Method for Solving Two Dimenional Heat conduction Equation within ocal Fractional Operator, Journal of Advance in Mathematic, 9 (24, 4, pp [2] Jafari, H., et al., Application of the ocal Fractional Adomian Decompoition and Serie Epanion Method for Solving Telegraph Equation on Cantor Set Involving ocal Fractional Derivative Operator, Journal of Zanko Sulaimani-Part A, 7 (25, June, pp.5-22 [24] Baleanu, D., et al., Approimate Analtical Solution of Gourat Problem within ocal Fractional Operator, Journal of Nonlinear Science and Application, 9 (26, 6, pp [25] Yang, X. J., et al., A New Numerical Technique for Solving the ocal Fractional Diffuion Equation: Two-Dimenional Etended Differential Tranform Approach, Applied Mathematic and Computation, 274 (26, Feb., pp. 4-5 [26] Jafari, H., et al., On the Approimate Solution of ocal Fractional Differential Equation with ocal Fractional Operator, Entrop, 8 (26, 5, pp. -2 [27] Jaim, H. K., On Approimate Method for Fractal Vehicular Traffic Flow, TWMS Journal of Applied and Engineering Mathematic, 7 (27,, pp [28] Srivatava, H. M., et al., ocal Fractional Sumudu Tranform with Application to IVP on Cantor Set, Abtract and Applied Anali, 24 (24, ID62529 [29] Jaim, H. K., The Analtical Solution for Volterra Integro-Differential Equation Involving ocal fractional Operator b Yang-aplace Tranform, Sahand Communication in Mathematical Anali, 6 (27,, pp [] Jafari, H., et al., Reduced Differential Tranform Method for Partial Differential Equation within ocal Fractional Derivative Operator, Advance in Mechanical Engineering, 8 (26, 4, pp. -6 [] Jaim, H. K., et al., ocal Fractional aplace Variational Iteration Method for Solving Diffuion and Wave Equation on Cantor Set within ocal Fractional Operator, Mathematical Problem in Engineering, 25 (25, ID987 [2] Jaim, H. K., The Approimate Solution of Three-Dimenional Diffuion and Wave Equation within ocal Fractional Derivative Operator, Abtract and Applied Anali, 26 (26, ID2959 Paper ubmitted: Augut 4, 27 Paper revied: November 2, 27 Paper accepted: November 24, Societ of Thermal Engineer of Serbia Publihed b the Vinča Intitute of Nuclear Science, Belgrade, Serbia. Thi i an open acce article ditributed under the CC BY-NC-ND 4. term and condition