A novel solution technique for two dimensional Burger’s equation

Alexandria Engineering Journal (2014) 53, 485–490 Alexandria University Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com ORIGINAL ARTICLE A novel solution technique for two dimensional Burger’s equation Majid Khan * Department of Sciences and Humanities, National University of Computer & Emerging Sciences, Islamabad, Pakistan Received 9 November 2013; revised 7 January 2014; accepted 16 January 2014 Available online 17 February 2014 KEYWORDS Laplace decomposition method; Two-dimensional Burger’s differential equations Abstract In this paper, the Laplace decomposition method (LDM) is proposed to solve the twodimensional nonlinear Burgers’ differential equations. Two test problems are considered to illustrate the accuracy of the proposed algorithm. It is shown that the numerical results are in good agreement with the exact solutions for each problem. ª 2014 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. 1. Introduction The Burger’s differential equation displays a sample model for describing the communication between reaction apparatuses, acoustic waves, convection effects, diffusion transports, heat conduction and modeling of dynamics. Burger’s prototype of turbulence is a very significant fluid dynamics archetype. The study of this model plays a vital role in understanding different numerical and analytical techniques. Also, this model provides a basis for theory of shock waves. Several authors have investigated Burger’s model for various physical flow problem in fluid dynamics. The mathematical structure of Burger’s equation was introduced by Johannes Martinus Burgers [1–5]. The general form of two-dimensional Burger’s equations is given as follows: * Fax: +92 0514100619. E-mail address: mk.cfd1@gmail.com. Peer review under responsibility of Faculty of Engineering, Alexandria University. Production and hosting by Elsevier   @u @u @u 1 @2u @2u ; þu þv ¼ þ @t @x @y Re @x2 @y2  2  @v @v @v 1 @ u @2u ; þu þv ¼ þ @t @x @y Re @x2 @y2 ð1Þ ð2Þ with subjected initial and boundary conditions are uðx; y;0Þ ¼ f1 ðx;yÞ; vðx;y;0Þ ¼ f2 ðx;yÞ; x;y 2 X; ð3Þ uðx; y;tÞ ¼ f3 ðx;y;tÞ; vðx; y;tÞ ¼ f4 ðx;y;tÞ; x;y 2 @X; t > 0; ð4Þ where uðx; y; tÞ and vðx; y; tÞ are velocity components in the direction of x and y-axes, Re is a Reynold number, fi i ¼ 1; 2 . . . 4. are given function on specified points, X ¼ fðx; yÞ j a 6 x 6 b; a 6 y 6 bg is a domian and @X is a boundary. Numerical schemes for the solutions of Burger’s equation normally categorize into the following classes: finite difference, finite element and spectral methods [6–8]. In [9], two dimensional Burger’s equation is solved with fully implicit finite-difference form. The analytical approximate solution schemes for nonlinear differential equations are of great significance in physical problems. Several analytical techniques such as Adomian decomposition method, homotopy analysis method, homotopy perturbation method and variational iterative 1110-0168 ª 2014 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. http://dx.doi.org/10.1016/j.aej.2014.01.004 486 M. Khan method are devised for the solution of Burger’s differential equation [10–16]. From last few years significant devotion has been given to Laplace decomposition method (LDM) and its modifications for resolving physical model equations [17–30]. To the best of our information, LDM is not so far applied to solve Burgers’ equation. Motivated by the work described in [31–40], we have used the LDM scheme to solve the two-dimensional Burgers’ equations. The structure of the article is as follows. In section 2, LDM is given. In Section 3, numerical applications of LDM are illustrated. The final remarks are given in the last section. 2. General mechannism of Laplace decomposition method u0 ðx; tÞ ¼ Gðx; tÞ; umþ1 ðx; tÞ ¼ L1 ð13Þ   1 ½L½A  þ L½Rðu Þ ; m P 0; m m sn where Gðx; tÞ represents the term arising from source term and prescribe initial conditions. The proposed method does not resort to linearization, assumptions of weak nonlinearity and it is more realistic compared to the method of simplifying the physical problems. 3. Numerical Implementation of Laplace decomposition method Example 1. Consider two dimensional nonlinear Burger’s differential equation [16] Consider equation Fðuðx; tÞÞ ¼ gðx; tÞ, where F represents a general nonlinear ordinary or partial differential operator including both linear and nonlinear terms. The linear terms are decomposes into L þ R, where L is the highest order linear operator and R is the remaining of the linear operator. Thus, the equation can be written as   @u @u @u 1 @2u @2u ; þu þv ¼ þ @t @x @y Re @x2 @y2  2  @v @v @v 1 @ v @2v þu þv ¼ þ ; @t @x @y Re @x2 @y2 Lu þ Ru þ Nu ¼ gðx; tÞ; with initial conditions ð5Þ where Nu, indicates the nonlinear terms. By applying Laplace transform on both sides of Eq. (5), we get L½Lu þ Ru þ Nu ¼ L½gðx; tÞ: n X sk1 uðnkÞ ðx; 0Þ þ L½Ru þ L½Nu ¼ L½gðx; tÞ: ð7Þ k¼1 Operating inverse Laplace transform on both sides of Eq. (7), we get   1 u ¼ Gðx; tÞ  L1 n ½L½Nu þ L½Ru : ð8Þ s The Laplace decomposition method assumes the solution u can be expanded into infinite series as 1 X um : u¼ ð9Þ m¼0 Also the nonlinear term Nu can be written as Nu ¼ 1 X Am ; ð10Þ m¼0 where Am are the Adomian polynomials [17]. By substituting Eqs. (9) and (10) in Eq. (8), the solution can be written as " " " " !### # 1 1 1 X X X 1 1 um Am þ L R L : um ðx; tÞ ¼ Gðx; tÞ  L sn m¼0 m¼0 m¼0 ð11Þ In Eq. (11), the Adomian polynomials can be generated by several means. Here we used the following recursive formulation: " !# 1 X 1 dm i Am ¼ ; m ¼ 0; 1; 2; . . . : ð12Þ N k u i m! dkm i¼0 k¼0 In general, the recursive relation is given by uðx; y; 0Þ ¼ x þ y; vðx; y; 0Þ ¼ x  y: ð15Þ ð16Þ ð17Þ Applying Laplace transform algorithm, we get ð6Þ Using the differential property of Laplace transform, we have sn L½u  ð14Þ     @u @u 1 þv ðr2 uÞ ; ¼L suðx; y; sÞ  uðx; y; 0Þ þ L u @x @y Re ð18Þ     @v @v 1 svðx; y; sÞ  vðx; y; 0Þ þ L u þv ðr2 vÞ ; ¼L @x @y Re ð19Þ uðx; y; sÞ ¼     uðx; y; 0Þ 1 @u @u 1 1  L u þv þ L ðr2 uÞ ; s s @x @y s Re ð20Þ vðx; y; sÞ ¼     vðx; y; 0Þ 1 @v @v 1 1  L u þv ðr2 vÞ : þ L s s @x @y s Re ð21Þ 2 2 @ @ where r2 ¼ @x 2 þ @y2 . Using given initial conditions Eqs. (20) and (21), becomes    
xþy 1 @u @u 1 1 ð22Þ  L u þv r2 u ; þ L uðx; y; sÞ ¼ s s @x @y s Re    
xy 1 @v @v 1 1 ð23Þ vðx; y; sÞ ¼  L u þv r2 v : þ L s s @x @y s Re Applying inverse Laplace transform to Eqs. (22) and (23), we have     
1 @u @u 1 1 uðx; y; tÞ ¼ x þ y  L1 L u þ L þv r2 u ; s @x @y s Re ð24Þ     
1 @v @v 1 1 2 L u þv rv : þ L vðx; y; tÞ ¼ x  y  L s @x @y s Re 1 ð25Þ The Laplace decomposition method (LDM) [1–3] assumes that the series solution of uðx; y; tÞ and vðx; y; tÞ are given by A novel solution technique for two dimensional Burger’s equation u¼ 1 X un ðx; y; tÞ; ð26Þ ð27Þ n¼0 Utilizing Eqs. (26) and (27), into Eqs. (24) and (25) yields: " " ## 1 1 1 X X X 1 1 L An ðuÞ þ Bn ðu; vÞ un ðx; y; tÞ ¼ x þ y  L s n¼0 n¼0 n¼0 " " !!## 1 X 1 1 ; ð28Þ un r2 þ L1 L s Re n¼0 " " ## 1 1 1 X X X 1 1 vn ðx; y; tÞ ¼ x  y  L Dn ðvÞ Cn ðu; vÞ þ L s n¼0 n¼0 n¼0 " " !!## 1 X 1 1 : ð29Þ vn r2 þ L1 L s Re n¼0 In above Eqs. (28) and (29), An ðuÞ; Bn ðu; vÞ, Cn ðu; vÞ and Dn ðvÞ are Adomian polynomials that represent nonlinear tearms given as follow: 1 X An ðuÞ ¼ uux ; n¼0 1 X Bn ðu; vÞ ¼ vuy ; From Eqs. (30)–(39), our required recursive relation is given by u0 ðx; y; tÞ ¼ x þ y; n¼0 1 X vn ðx; y; tÞ: v¼ 487 ð30Þ ð31Þ ð46Þ " " ## 1 1 X X 1 1 Bn ðu; vÞ An ðuÞ þ uðx; y; tÞ ¼ L L s n¼0 n¼0 " " !!## 1 X 1 1 1 2 þL L r un ; s Re n¼0 v0 ðx; y; tÞ ¼ x  y; ð48Þ " # 1 1 X 1 X Dn ðu; vÞ Cn ðu; vÞ þ L vðx; y; tÞ ¼ L s n¼0 n¼0 " " !!## 1 X 1 1 1 2 vn : L r þL s Re n¼0 1 u1 ðx; y; tÞ ¼ 2xt; v1 ðx; y; tÞ ¼ 2yt; ð50Þ ð51Þ u2 ðx; y; tÞ ¼ 2xt2 þ 2yt2 ; ð52Þ v2 ðx; y; tÞ ¼ 2xt2  2yt2 ; ð53Þ u3 ðx; y; tÞ ¼ 4yt4  4xt3 ; ð54Þ 3 v3 ðx; y; tÞ ¼ 4xt  4yt : Cn ðu; vÞ ¼ uvx ; n¼0 1 X Dn ðu; vÞ ¼ uvy : ð32Þ ð33Þ n¼0 ð49Þ The first few components of un ðx; y; tÞ and vn ðx; y; tÞ follows immediately upon setting Re ¼ 1, we have 4 n¼0 1 X ð47Þ ð55Þ Therefore solution obtained by LDM is given below: 1 X un ðx; y; tÞ uðx; y; tÞ ¼ n¼0 ¼ x þ y  2xt þ 2xt2 þ 2yt2 þ 4yt4  4xt3 þ    The few components of the Adomian polynomials are given below: A0 ðuÞ ¼ u0x ; A1 ðuÞ ¼ 2u20x u1x ; .. . n X An ðuÞ ¼ uix uðniÞx ; ð36Þ ð37Þ ð38Þ ð39Þ ð40Þ ð41Þ ð42Þ i¼0 D0 ðu; vÞ ¼ u0 v0y ; D1 ðu; vÞ ¼ u0 v1y þ u1 v0y ; .. . n X ui vðniÞy : Dn ðu; vÞ ¼ i¼0 ¼ ðx  2xt þ yÞð1 þ 2t2 þ 4t4 þ . . .Þ; x  2xt þ y : ¼ 1  2t2 vðx; y; tÞ ¼ 1 X ð56Þ vn ðx;y; tÞ ¼ x  y  2yt þ 2xt2  2yt2 þ 4xt4  4yt3 þ . . . n¼0 i¼0 C0 ðu; vÞ ¼ u0 v0x ; C1 ðu; vÞ ¼ u0 u1x þ u1 v0x ; .. . n X ui vðniÞx ; Cn ðu; vÞ ¼ þ yð1 þ 2y2 þ 4t4 þ   Þ; ð34Þ ð35Þ i¼0 B0 ðuÞ ¼ v0 u0y ; B1 ðuÞ ¼ v0 u1y þ v1 u0y ; .. . n X vi uðniÞx ; Bn ðu; vÞ ¼ ¼ xð1 þ 2t2 þ 4t4 þ . . .Þ  2xtð1 þ 2t2 þ   Þ ð43Þ ð44Þ ð45Þ ¼ xð1 þ 2t2 þ 4t4 þ . . .Þ  2ytð1 þ 2t2 þ . . .Þ  yð1 þ 2y2 þ 4t4 þ . ..Þ; ¼ ðx  2yt  yÞð1 þ 2t2 þ 4t4 þ . . .Þ; x  2yt  y ¼ : 1  2t2 ð57Þ which is the exact solution of two dimensional Burger’s equations [16]. Example 2. Let us consider another system of two dimensional Burger’s equations with the following initial conditions 3 1 uðx; y; 0Þ ¼  ; vðx; y; 0Þ 4 4ð1 þ eReðxþyÞ=8 Þ 3 1 : ð58Þ ¼ þ 4 4ð1 þ eReðxþyÞ=8 Þ The exact solution of problem is 3 1 ; uðx; y; tÞ ¼  4 4ð1 þ eReðt4xþ4yÞ=32 Þ 3 1 vðx; y; tÞ ¼ þ : 4 4ð1 þ eReðt4xþ4yÞ=32 Þ ð59Þ ð60Þ 488 Table 1 M. Khan Comparison of LDM solution (Eight order solution) with exact solution for Re ¼ 1, with mesh points x ¼ 0:1 and y ¼ 0:1. t Exact uðx; y; tÞ LDM uðx; y; tÞ Exact vðx; y; tÞ LDMvðx; y; tÞ 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.190955 0.183673 0.178010 0.173913 0.171429 0.170732 0.172185 0.176471 0.184874 0.200000 0.190955 0.183673 0.178010 0.173913 0.171429 0.170734 0.172187 0.176473 0.184876 0.200000 0.0100503 0.0204082 0.0314136 0.0434783 0.0571429 0.0731707 0.0927152 0.117647 0.151261 0.200000 0.0100503 0.0204082 0.0314136 0.0434783 0.0571429 0.0731709 0.0927154 0.117647 0.151264 0.200000 Table 2 Comparison of LDM solution (Eight order solution) with exact solution for Re ¼ 1, with mesh points x ¼ 0:3 and y ¼ 0:1. t Exact uðx; y; tÞ LDM uðx; y; tÞ Exact vðx; y; tÞ LDM vðx; y; tÞ 0.05 0.1 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.371859 0.346939 0.324607 0.304348 0.285714 0.268293 0.251656 0.235294 0.218487 0.200000 0.371859 0.346939 0.324607 0.304348 0.285715 0.268295 0.251659 0.235296 0.218488 0.200000 0.190955 0.183673 0.178012 0.173913 0.171429 0.170732 0.172184 0.176470 0.184873 0.200000 0.190955 0.183673 0.178012 0.173913 0.171430 0.170734 0.172185 0.176471 0.184874 0.200000 Table 3 Comparison of LDM solution (Eight order solution) with exact solution for Re ¼ 1, with mesh points x ¼ 0:1 and y ¼ 0:3. t Exact uðx; y; tÞ LDM uðx; y; tÞ Exact vðx; y; tÞ LDM vðx; y; tÞ 0.05 0.1 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.39196 0.387755 0.387435 0.391304 0.400000 0.414634 0.437086 0.470588 0.521008 0.600000 0.39196 0.387755 0.387435 0.391304 0.400000 0.414636 0.437088 0.470589 0.521009 0.600000 0.231156 0.265306 0.303665 0.347826 0.400000 0.463415 0.543046 0.647059 0.789916 1.000000 0.231156 0.265306 0.303665 0.347827 0.400011 0.463417 0.543048 0.647061 0.789919 1.000000 Table 4 Comparison of LDM solution (Eight order solution) with exact solution for Re ¼ 1, with mesh points x ¼ 0:1 and y ¼ 0:5. t Exact uðx; y; tÞ LDM uðx; y; tÞ Exact vðx; y; tÞ LDM vðx; y; tÞ 0.05 0.1 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.532084 0.530712 0.529391 0.528118 0.526894 0.525717 0.524587 0.523511 0.522457 0.521456 0.532084 0.530712 0.529391 0.528118 0.526894 0.525717 0.524587 0.523511 0.522457 0.521456 0.967916 0.969288 0.97061 0.971882 0.973106 0.974283 0.975413 0.9765 0.977543 0.978544 0.967916 0.969288 0.97061 0.971882 0.973106 0.974283 0.975413 0.9765 0.977543 0.978544 A novel solution technique for two dimensional Burger’s equation Table 5 489 Comparison of LDM solution (Eight order solution) with exact solution for Re ¼ 1, with mesh points x ¼ 0:3 and y ¼ 0:5. t Exact uðx; y; tÞ LDM uðx; y; tÞ Exact vðx; y; tÞ LDM vðx; y; tÞ 0.05 0.1 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.515513 0.514801 0.514120 0.513468 0.512845 0.512250 0.511680 0.511136 0.510616 0.510119 0.515513 0.514801 0.514120 0.513468 0.512845 0.512250 0.511680 0.511136 0.510616 0.510119 0.984487 0.985199 0.985880 0.986532 0.987155 0.987750 0.988320 0.989864 0.989384 0.989881 0.984487 0.985199 0.985880 0.986532 0.987155 0.987750 0.988320 0.989864 0.989384 0.989881 Table 6 Comparison of LDM solution (Eight order solution) with exact solution for Re ¼ 1, with mesh points x ¼ 0:1 and y ¼ 0:9. t Exact uðx; y; tÞ LDM uðx; y; tÞ Exact vðx; y; tÞ LDM vðx; y; tÞ 0.05 0.1 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.605429 0.602393 0.599384 0.596406 0.593462 0.590555 0.587688 0.584863 0.582083 0.579351 0.605429 0.602393 0.599384 0.596406 0.593462 0.590555 0.587688 0.584863 0.582083 0.579351 0.894571 0.897607 0.900616 0.903594 0.906538 0.909445 0.912312 0.915137 0.917917 0.920649 0.894571 0.897607 0.900616 0.903594 0.906538 0.909445 0.912312 0.915137 0.917917 0.920649 The few components of series solution obtained with the help of Laplace decomposition method are as follow: u1 ðx; y; tÞ ¼  þ v1 ðx; y; tÞ ¼ exþy Ret 64 1 þ 18 exþy Re exþy t 16 1 þ 18 exþy Re exþy Ret 64 1 þ 
3 
3 1 xþy e Re 8 xþy e
3 ; þ t 16ð1 þ 18 exþy ReÞ 3 e2xþ2y Ret 64 1 þ 18 exþy Re References ð61Þ e2xþ2y Ret 64 1 þ 18 exþy Re ;
3
3 non-linear equations arises in physical sciences and does not require linearization, discretization or perturbation and occupy less memory space in execution of a recursive relation. ð62Þ The efficiency of Laplace decomposition method for system of two dimensional Burger’s equations for above two examples have a closed agreement with exact solution. The comparison between Laplace decomposition method with exact solution is listed in Tables 1–6 for different values of Renoyld number in case of above examples. The numerical results show that the LDM may serve as a replacement to the solution of nonlinear problems in physical sciences. 4. Conclusion The aim here is to provide the exact and series solution of a Burger’s equation by using Laplace decomposition method (LDM). The convergence of LDM is also shown in Tables 1–6. The results of LDM are compared with exact solution. The results of LDM have a closed agreement with exact solution. The analysis given here shows further confidence on LDM. Therefore, this method can be applied to other [1] H. Bateman, Some recent researches on the motion of fluids, Monthly Weather Rev. 43 (1915) 163–170. [2] J.M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948) 171–199. [3] P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, 1973. [4] J. Smoller, Shock Waves and Reaction–Diffusion Equations, Springer-Verlag, 1983. [5] E. Hopf, The partial differential equation ut þ uuxx ¼ luxx , Comm. Pure Appl. Math. 3 (1950) 201–230. [6] M. Basto, V. Semiao, F. Calheiros, Dynamics and synchronization of numerical solutions of the Burgers equation, J. Comput. Appl. Math. 231 (2009) 793–806. [7] A.H.A. Ali, G.A. Gardner, A collocation solution for Burgers’ equation using cubic B-spline finite elements, Comput. Meth. Appl. Mech. Eng. 100 (1992) 325–337. [8] W.M. Moslem, R. Sabry, Zakharov–Kuznetsov–Burgers equation for dust ion acoustic waves, Chaos Soliton. Fract. 36 (2008) 628–634. [9] W. Liao, An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation, Appl. Math. Comput. 206 (2008) 755–764. [10] A.R. Bahadir, A fully implicit finite-difference scheme for twodimensional Burgers’ equations, Appl. Math. Comput. 137 (2003) 131–137. [11] A. Alharbi, E.S. Fahmy, ADM-Padé solutions for generalized Burgers and Burgers–Huxley systems with two coupled equations, J. Comput. Appl. Math. 233 (2010) 2071–2080. 490 [12] A. Gorguis, A comparison between Cole–Hopf transformation and the decomposition method for solving Burgers’ equations, Appl. Math. Comput. 173 (2006) 126–136. [13] M. Dehghan, A. Hamidi, M. Shakourifar, The solution of coupled Burgers’ equations using Adomian–Padé technique, Appl. Math. Comput. 189 (2007) 1034–1047. [14] M. Inc, On numerical solution of Burgers’ equation by homotopy analysis method, Phys. Lett. A 372 (2008) 356–360. [15] A. Molabahrami, F. Khani, S. Hamedi-Nezhad, Soliton solutions of the two-dimensional KdV–Burgers equation by homotopy perturbation method, Phys. Lett. A 370 (2007) 433– 436. [16] J. Biazar, H. Aminikhah, Exact and numerical solutions for non-linear Burger’s equation by VIM, Math. Comput. Model. 49 (2009) 1394–1400. [17] M. Hussain, M. Khan, Modified Laplace decomposition method, Appl. Math. Sci. 4 (2010) 1769–1783. [18] M. Khan, M. Hussain, Application of Laplace decomposition method on semi infinite domain, Numer. Algorithms 56 (2011) 211–218. [19] M.A. Gondal, M. Khan, Homotopy perturbation method for nonlinear exponential boundary layer equation using laplace transformation, He’s polynomials and pade technology He’s polynomials and pade technology, Int. J. Nonlinear Sci. Numer. 12 (2010) 1145–1153. [20] M. Khan, M.A. Gondal, An efficient two step Laplace decomposition algorithm for singular Volterra integral equations, Int. J. Phys. Sci. 6 (2011) 4717–4720. [21] M. Khan, M.A. Gondal, A new analytical approach to solve Thomas–Fermi equation, World Appl. Sci. J. 12 (2011) 2309– 2313. [22] M. Khan, M.A. Gondal, A reliable treatment of Abel’s second kind singular integral equations, Appl. Math. Lett. 25 (2012) 1666–1670. [23] M. Khan, M.A. Gondal, S. Kumar, A new analytical solution procedure for nonlinear integral equations, Math. Comput. Model. 55 (2012) 1892–1897. [24] M. Khan, M.A. Gondal, New computational dynamics for magnetohydrodynamics flow over a nonlinear stretching sheet, Zeit. Naturforsch. 67a (2012) 262–266. [25] M. Khan, M.A. Gondal, S.I. Batool, A novel analytical implementation of nonlinear volterra integral equations, Zeit. Naturforsch. 67a (2012) 674–678. [26] S. Salahshour, M. Khan, Exact solutions of nonlinear interval Volterra integral equations, Int. J. Ind. Math. 4 (2012) 375–388. M. Khan [27] M. Khan, M.A. Gondal, K. Omrani, A new analytical approach to two dimensional magneto-hydrodynamics flow over a nonlinear porous stretching sheet by Laplace Padé decomposition method, Results Math. 63 (2013) 289–301. [28] M.A. Gondal, S.I. Batool, M. Khan, A novel fractional Laplace decomposition method for chaotic systems and the generation of chaotic sequences, J. Vib. Control. http://dx.doi.org/10.1177/ 1077546312475149. [29] M. Khan, M.A. Gondal, S.I. Batool, A new modified Laplace decomposition method for higher order boundary value problems, Comput. Math. Organ Theory 19 (2013) 446–459. [30] M. Khan, A new algorithm for higher order integro-differential equations, Afr. Matemat. (2013), http://dx.doi.org/10.1007/ s13370-013-0200-4. [31] A. Salah, M. Khan, M.A. Gondal, A novel solution procedure for fuzzy fractional heat equations by homotopy analysis transform method, Neural Comput. Appl. 23 (2013) 269–271. [32] M.A. Godal, A. Salah, M. Khan, S.I. Batool, A novel analytical solution of a fractional diffusion problem by homotopy analysis transform method, Neural Comput. Appl. 23 (2013) 1643–1647. [33] M. Khan, M.A. Gondal, A new analytical solution of foam drainage equation by Laplace decomposition method, Adv. Res. Different. Eqs. 2 (2010) 53–64. [34] M. Khan, M.A. Gondal, S. Kumar, A novel homotopy perturbation transform algorithm for linear and nonlinear system of partial differential equations, World Appl. Sci. J. 12 (2011) 2352–2357. [35] Sunil Kumar, A fractional model of gas dynamics equation by using Laplace transform, Zeit. Naturforsch. 67a (2012) 389–396. [36] Sunil Kumar, W. Leilei, A fractional model of diffusion equation by using Laplace transform, Sci. Irant. 19 (4) (2012) 1117–1123. [37] Sunil Kumar, Jagdev Singh, Devendra Kumar, New treatment of fractional Fornberg Whitham equation via Laplace transform, Ain Sham Eng. J. 4 (2013) 557–562. [38] Sunil Kumar, H. Jafari, K. Sayevand, L. Wei, A analytical solution of black-scholes option pricing equation by using Laplace transform, J. Fract. Cal. Appl. 2 (8) (2012) 1–9. [39] S.R. Ananth Kumar, R. Rangarajan, Application of Laplace decomposition method for solving certain differential difference equation of order (1,2), Bulletin of International Mathematical Virtual Institute 3 (2013) 103–111. [40] Fu-Kang Yin, Wang-Yi Han, Jun-Qiang Song, Modified Laplace decomposition method for Lane–Emden type differential equations, Int. J. Appl. Phys. Math. 3 (2013) 98–102.