Variational iteration method for solving multispecies Lotka Volterra equations
|
|
- Abraham Richardson
- 5 years ago
- Views:
Transcription
1 Computers and Mathematics with Applications Variational iteration method for solving multispecies Lotka Volterra equations B. Batiha, M.S.M. Noorani, I. Hashim School of Mathematical Sciences, National University of Malaysia, 436 UKM Bangi Selangor, Malaysia Received 28 September 26; accepted 2 December 26 Abstract This paper applies the variational iteration method to multispecies Lotka Volterra equations. Comparisons with the Adomian decomposition and the fourth-order Runge Kutta methods show that the variational iteration method is a powerful method for nonlinear equations. c 27 Elsevier Ltd. All rights reserved. Keywords: Variational iteration method; Adomian decomposition method; Fourth-order Runge Kutta method; Lotka Volterra equations 1. Introduction The Lotka Volterra equations model the dynamic behaviour of an arbitrary number of competitors [1]. Though originally formulated to describe the time history of a biological system, these equations find their application in a number of engineering fields such as simultaneous chemical and nonlinear control. In fact, the one-predator one-prey Lotka Volterra model is one of the most popular ones to demonstrate a simple nonlinear control system. The accurate solutions of the Lotka Volterra equations may become a difficult task either if the equations are stiff even with a small number of species, or when the number of species is large [2]. Unlike the discrete solutions obtained by the purely numerical methods like the fourth-order Runge Kutta method RK4, approximate analytical solutions can increase our insights into the natural behaviour of complex systems. An analytical method called the Adomian decomposition method ADM proposed by Adomian [3] aims to solve frontier nonlinear physical problems. It has been applied to a wide class of deterministic and stochastic problems, linear and nonlinear, in physics, biology and chemical reactions etc. For nonlinear models, the method has shown reliable results in supplying analytical approximations that converge rapidly [4]. Yet another powerful analytical method for nonlinear equations is called the variational iteration method VIM, which was first envisioned by He [5] modifying the approach by Inokuti et al. [6]. VIM has successfully been applied to many situations. For example, He [7] solved the classical Blasius equation using VIM. In [8], He gave a solution for a seepage flow problem with fractional derivatives in porous media using VIM. He [9] also employed VIM to Corresponding author. Tel.: ; fax: address: msn@ukm.my M.S.M. Noorani /$ - see front matter c 27 Elsevier Ltd. All rights reserved. doi:1.116/j.camwa
2 94 B. Batiha et al. / Computers and Mathematics with Applications give approximate solutions for some well-known nonlinear problems and in [1], a successful application of VIM to autonomous systems of ordinary differential equations is shown. VIM was also demonstrated to be a powerful method for strongly nonlinear equations by He [11 14]. Many other researchers have shown further applications of VIM. For example, Soliman [15] applied VIM to solve the KdV Burger s and Lax s seventh-order KdV equations. For the application of VIM to other Burger s related equations, see [16,17]. VIM has recently been applied to the solution of nonlinear coagulation problem with mass loss by Abulwafa et al. [18]. Momani et al. [19] applied VIM to the Helmholtz equation. VIM has been applied for solving nonlinear fractional differential equations by Odibat et al. [2]. Bildik et al. [21] used VIM for solving different types of nonlinear partial differential equations. In this paper, we apply VIM to the nonlinear multispecies Lotka Volterra equations. Comparisons with ADM and RK4 shall be made to determine the performance of VIM. 2. Variational iteration method The main feature of the method is that the solution of a mathematical problem with linearization assumption is used as initial approximation or trial function, then a more highly precise approximation at some special point can be obtained. This approximation converges rapidly to an accurate solution. To illustrate the basic concepts of VIM, we consider the following nonlinear differential equation: Lu + Nu = gx, where L is a linear operator, N is a nonlinear operator, and gx is an inhomogeneous term. According to VIM [9 14], we can construct a correction functional as follows: u n+1 x = u n x + x λ{lu n τ + Nũ n τ gτ}dτ, 2 where λ is a general Lagrangian multiplier [6], which can be identified optimally via the variational theory, the subscript n denotes the nth-order approximation, ũ n is considered as a restricted variation [9 11], i.e. δũ n =. 3. Analysis of multispecies Lotka Volterra equations Consider the general Lotka Volterra model for an m species system given as dn i m = N i b i + a i j N j, i = 1, 2,..., m. 3 j=1 These equations may represent either predator prey or competition cases One species In the one-species case, Eq. 3 reduces to one species competing for a given finite source of food, dn = Nb + an, b >, a <, N >, 4 where a and b are constants. This equation has an exact solution be bt, for b, b+an Nt = N aebt N, for b =, 1 ant where N is the initial condition. Solving Eq. 4 by ADM [3] yields the following recursive algorithm, N = N, N n+1 t = bnn + a A 1,n, n, 6 1 5
3 B. Batiha et al. / Computers and Mathematics with Applications where the Adomian polynomials A 1,n are given by A 1,n = N k N n k. Now, formally in VIM [5], we construct the correction functional, [ ] dnn s N n+1 t = N n t + λs bn n s añn 2 ds s ds, 8 where Ñ n is considered as restricted variations, i.e. δñ n =. Its stationary conditions can be obtained as 1 + λt =, λ s + bλs s=t =. The Lagrange multiplier λ can therefore be identified as λs = e bs t and the following variational iteration formula is obtained, e bs [ ] dnn s N n+1 t = N n t e bt bn n s ann 2 ds s ds, n. 1 The solution of the linearized version of Eq. 4, where a =, is Nt = Ce bt. Taking this as the initial approximation N gives 7 9 N 1 t = Ce bt + ac2 e bt e bt 1. b The condition N =.1 gives us C =.1. Thus 11 N 1 t =.1e bt +.1aebt e bt 1. b In the same manner, the rest of the components of the iteration formula 1 can be obtained using the computer algebra package Maple Two species The Lotka Volterra equations modelling two species competing for a common ecological niche are dn 1 = N 1 b 1 + a 11 N 1 + a 12 N 2, dn 2 = N 2 b 2 + a 21 N 1 + a 22 N 2, where a 11, a 12, a 21, a 22, b 1 and b 2 are constants. The Adomian recursive algorithms for solving 13 and 14 are, cf. [3], N 1, = N 1, N 2, = N 2, N 1,n+1 = N 2,n+1 = b1 N 1,n + a 11 A 1,n + a 12 A 1,2,n, n, 16 b2 N 2,n + a 21 A 2,1,n + a 22 A 2,n, n, 17 where N 1 and N 2 are the initial conditions and the Adomian polynomials are given by A 1,n = N 1,k N 1,n k, A 1,2,n = N 1,k N 2,n k, 18
4 96 B. Batiha et al. / Computers and Mathematics with Applications A 2,n = N 2,k N 2,n k, A 2,1,n = N 2,k N 1,n k. 19 The correction functionals for system 13 and 14 are [ ] dn1,n s N 1,n+1 t = N 1,n t + λ 1 s b 1 N 1,n s a 11 Ñ1,n 2 ds s a 12Ñ1,nsÑ 2,n s ds, 2 [ ] dn2,n s N 2,n+1 t = N 2,n t + λ 2 s b 2 N 2,n s a 22 Ñ2,n 2 ds s a 21Ñ2,nsÑ 1,n s ds, 21 where Ñ i,n are considered as restricted variations, i.e. δñ i,n =. Its stationary conditions can be obtained as 1 + λ 1 t =, λ 1 s + λ 1s s=t =, λ 2 t =, λ 2 s + λ 2s s=t =. 23 The Lagrange multipliers, therefore, can be identified as λ 1 s = e b 1s t and λ 2 s = e b 2s t. The solutions of the linearized versions of 13 and 14, where a 11 =, a 12 =, a 21 = and a 22 =, are N 1 t = C 1 e b 1t and N 2 t = C 2 e b 2t. Now taking these as initial estimates and imposing the conditions N 1 = 4 and N 2 = 1, for example, give C 1 = 4 and C 2 = 1. Thus the first iteration solutions are given by N 1,1 t = 4e b 1t + 4eb 1t 4a 11 b 2 1a 12 b 1 + 4a 11 b 2 e b 1t + 1a 12 b 1 e b 2t b 1 b 2, 24 N 2,1 t = 1e b 2t + 1eb 2t 4a 21 b 2 1a 22 b 1 + 4a 21 b 2 e b 1t + 1a 22 b 1 e b 2t b 1 b Again, the rest of the components of the iteration formulas 2 and 21 can be obtained using the computer algebra package Maple Three species The following version of the Lotka Volterra equations modelling three species shall be used [22]: dn 1 = N 1 1 N 1 αn 2 β N 3, dn 2 = N 2 1 β N 1 N 2 αn 3, dn 3 = N 3 1 αn 1 β N 2 N 3, where α and β are constants. The Adomian recursive algorithms for solving are, cf. [3], N 1, = N 1, N 2, = N 2, N 3, = N 3, 29 N 1,n+1 = N 2,n+1 = N 3,n+1 = N1,n A 1,n α A 1,2,n β A 1,3,n, n, 3 N2,n β A 2,1,n A 2,n α A 2,3,n, n, 31 N3,n α A 3,1,n β A 3,2,n A 3,n, n, 32 where N 1, N 2 and N 3 are the initial conditions and the Adomian polynomials are A 1,n = N 1,k N 1,n k, A 1,2,n = N 1,k N 2,n k, A 1,3,n = N 1,k N 3,n k, 33
5 B. Batiha et al. / Computers and Mathematics with Applications Table 1 Numerical comparisons when b = 1, a = 3, N =.1 t Exact solution ADM, φ 3 2-iteration VIM A 2,n = A 3,n = N 2,k N 2,n k, A 2,1,n = N 3,k N 3,n k, A 3,1,n = In VIM, the correction functionals are N 1,n+1 t = N 1,n t + N 2,n+1 t = N 2,n t + N 3,n+1 t = N 3,n t + λ 1 s N 2,k N 1,n k, A 2,3,n = N 3,k N 1,n k, A 3,2,n = [ dn1,n ds [ dn2,n λ 2 s ds N 2,k N 3,n k, 34 N 3,k N 2,n k. 35 ] N 1,n + Ñ1,n 2 + αñ 1,n Ñ 2,n + β Ñ 1,n Ñ 3,n ds, 36 ] N 2,n + β Ñ 2,n Ñ 1,n + Ñ2,n 2 + αñ 2,n Ñ 3,n ds, 37 [ dn3,n λ 3 s N 3,n + αñ 3,n Ñ 1,n + β Ñ 3,n Ñ 2,n + Ñ3,n 2 ds ] ds, 38 where Ñ i,n are considered as restricted variations, i.e. δñ i,n =. Its stationary conditions can be obtained as 1 + λ 1 t =, λ 1 s + λ 1s s=t =, λ 2 t =, λ 2 s + λ 2s s=t =, λ 3 t =, λ 3 s + λ 3s s=t =. 41 Thus, the Lagrange multipliers are λ 1 s = λ 2 s = λ 3 s = e s+t. The solutions of the linearized versions of are N 1 t = C 1 e t, N 2 t = C 2 e t and N 3 t = C 3 e t. Now taking these as initial estimates and imposing the conditions N 1 =.2, N 2 =.3 and N 3 =.5, for example, give C 1 =.2, C 2 =.3 and C 3 =.5. Thus, the first iteration solutions are N 1,1 t =.2e t +.4e t +.6αe t +.1βe t.4e 2t.6αe 2t.6βe 2t, 42 N 2,1 t =.3e t +.6βe t +.9e t +.15αe t.6βe 2t.9e 2t.15αe 2t, 43 N 3,1 t =.3e t +.6αe t +.15βe t +.9e t.6αe 2t.15βe 2t.25e 2t. 44 Again, the next iterations can be obtained using the computer algebra package Maple. 4. Numerical results and discussion The numerical solutions obtained by using the VIM are compared with the exact solution for the one-species case, and those obtained by ADM and RK4. Table 1 shows comparison between the 2-iteration of VIM, 3-term ADM and the exact solution for the one species in the case b = 1, a = 3 and N =.1. The results show the good accuracy
6 98 B. Batiha et al. / Computers and Mathematics with Applications Table 2 Numerical comparisons in the case b 1 =.1, a 11 =.14, a 12 =.12, b 2 =.8, a 21 =.9, a 22 =.1, N 1 = 4 and N 2 = 1 t ADM, φ 3 2-iteration VIM RK4, h =.1 N 1 N 2 N 1 N 2 N 1 N Table 3 Numerical comparisons when α =.1, β =.1, N 1 =.2, N 2 =.3, N 3 =.5 t ADM, φ 3 4-iteration VIM RK4, h =.1 N 1 N 2 N 3 N 1 N 2 N 3 N 1 N 2 N of VIM. In Table 2 we show the comparison between the 2-iteration of VIM, 3-term ADM and RK4 solutions for the two species in the case b 1 =.1, a 11 =.14, a 12 =.12, b 2 =.8, a 21 =.9, a 22 =.1, N 1 = 4 and N 2 = 1. Clearly, high accuracy is achieved with only two iterations of VIM. The numerical solutions for the three-species case are tabulated in Table 3 in the case α =.1, β =.1, N 1 =.2, N 2 =.3 and N 3 =.5. Again, the numerical results show that VIM is of high accuracy. 5. Conclusions In this paper, the VIM is applied to the solution of nonlinear multispecies Lotka Volterra equations. Comparisons with the Adomian decomposition method and the fourth-order Runge Kutta method show that the VIM is a powerful method for nonlinear equations. The advantage of the VIM over the ADM is that there is no need for the evaluations of the Adomian polynomials and the advantage over the RK4 method is that VIM or variational iteration method gives continuous solutions. Acknowledgement The financial support received from the Academy of Sciences Malaysia under the SAGA grant no. P24c is gratefully acknowledged. References [1] J. Hofbauer, K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, London, 1988.
7 B. Batiha et al. / Computers and Mathematics with Applications [2] S. Olek, An accurate solution to the multispecies Lotka Volterra equations, SIAM Rev [3] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Boston, [4] W. Chen, Z. Lu, An algorithm for Adomian decomposition method, Appl. Math. Comput [5] J.H. He, A new approach to nonlinear partial differential equations, Commun. Nonlinear Sci. Numer. Simul [6] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: S. Nemat-Nassed Ed., Variational Method in the Mechanics of Solids, Pergamon Press, 1978, pp [7] J.H. He, Approximate analytical solution of Blasius equation, Commun. Nonlinear Sci. Numer. Simul [8] J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg [9] J.H. He, Variational iteration method a kind of non-linear analytical technique: some examples, Int. J. Non-Linear Mech [1] J.H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput [11] J.H. He, Y.Q. Wan, Q. Guo, An iteration formulation for normalized diode characteristics, Int. J. Circuit Theory Appl [12] J.H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Modern Phys. B [13] J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation, de-verlag im Internet GmbH, Berlin, 26. [14] J.H. He, Variational iteration method Some recent results and new interpretations. J. Comput. Appl. Math. in press. [15] A.A Soliman, Numerical simulation and explicit solutions of KdV Burgers and Lax s seventh-order KdV equations, Chaos Solitons Fractals [16] M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burger s and coupled Burger s equations, J. Comput. Appl. Math [17] M. Moghimi, F.S.A. Hejazi, Variational iteration method for solving generalized Burgers Fisher and Burgers equations, Chaos Solitons Fractals [18] E.M. Abulwafa, M.A. Abdou, A.A. Mahmoud, The solution of nonlinear coagulation problem with mass loss, Chaos Solitons Fractals [19] S. Momani, S. Abuasad, Application of He s variational iteration method to Helmholtz equation, Chaos Solitons Fractals [2] Z.M. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul [21] N. Bildik, A. Konuralp, The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, Int. J. Nonlinear Sci. Numer. Simul [22] R.M. May, W.J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math
Application of Variational Iteration Method to a General Riccati Equation
International Mathematical Forum,, 007, no. 56, 759-770 Application of Variational Iteration Method to a General Riccati Equation B. Batiha, M. S. M. Noorani and I. Hashim School of Mathematical Sciences
More informationAn Analytic Study of the (2 + 1)-Dimensional Potential Kadomtsev-Petviashvili Equation
Adv. Theor. Appl. Mech., Vol. 3, 21, no. 11, 513-52 An Analytic Study of the (2 + 1)-Dimensional Potential Kadomtsev-Petviashvili Equation B. Batiha and K. Batiha Department of Mathematics, Faculty of
More informationComputers and Mathematics with Applications. A new application of He s variational iteration method for the solution of the one-phase Stefan problem
Computers and Mathematics with Applications 58 (29) 2489 2494 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa A new
More informationThe Modified Variational Iteration Method for Solving Linear and Nonlinear Ordinary Differential Equations
Australian Journal of Basic and Applied Sciences, 5(10): 406-416, 2011 ISSN 1991-8178 The Modified Variational Iteration Method for Solving Linear and Nonlinear Ordinary Differential Equations 1 M.A. Fariborzi
More informationVariational iteration method for fractional heat- and wave-like equations
Nonlinear Analysis: Real World Applications 1 (29 1854 1869 www.elsevier.com/locate/nonrwa Variational iteration method for fractional heat- and wave-like equations Yulita Molliq R, M.S.M. Noorani, I.
More informationVariational Iteration Method for a Class of Nonlinear Differential Equations
Int J Contemp Math Sciences, Vol 5, 21, no 37, 1819-1826 Variational Iteration Method for a Class of Nonlinear Differential Equations Onur Kıymaz Ahi Evran Uni, Dept of Mathematics, 42 Kırşehir, Turkey
More informationNumerical Simulation of the Generalized Hirota-Satsuma Coupled KdV Equations by Variational Iteration Method
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.7(29) No.1,pp.67-74 Numerical Simulation of the Generalized Hirota-Satsuma Coupled KdV Equations by Variational
More informationComputers and Mathematics with Applications. A modified variational iteration method for solving Riccati differential equations
Computers and Mathematics with Applications 6 (21) 1868 1872 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa A modified
More informationVariational Iteration Method for Solving Nonlinear Coupled Equations in 2-Dimensional Space in Fluid Mechanics
Int J Contemp Math Sciences Vol 7 212 no 37 1839-1852 Variational Iteration Method for Solving Nonlinear Coupled Equations in 2-Dimensional Space in Fluid Mechanics A A Hemeda Department of Mathematics
More informationA Variational Iterative Method for Solving the Linear and Nonlinear Klein-Gordon Equations
Applied Mathematical Sciences, Vol. 4, 21, no. 39, 1931-194 A Variational Iterative Method for Solving the Linear and Nonlinear Klein-Gordon Equations M. Hussain and Majid Khan Department of Sciences and
More informationA Multistage Adomian Decomposition Method for Solving The Autonomous Van Der Pol System
Australian Journal of Basic Applied Sciences, 3(4): 4397-4407, 2009 ISSN 1991-8178 A Multistage Adomian Decomposition Method for Solving The Autonomous Van Der Pol System Dr. Abbas Y. AL_ Bayati Dr. Ann
More informationA Study of the Variational Iteration Method for Solving. Three Species Food Web Model
Int. Journal of Math. Analysis, Vol. 6, 2012, no. 16, 753-759 A Study of the Variational Iteration Method for Solving Three Species Food Web Model D. Venu Gopala Rao Home: Plot No.159, Sector-12, M.V.P.Colony,
More informationSolving the Fisher s Equation by Means of Variational Iteration Method
Int. J. Contemp. Math. Sciences, Vol. 4, 29, no. 7, 343-348 Solving the Fisher s Equation by Means of Variational Iteration Method M. Matinfar 1 and M. Ghanbari 1 Department of Mathematics, University
More informationComputers and Mathematics with Applications
Computers and Mathematics with Applications 58 (29) 27 26 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Study on
More informationResearch Article He s Variational Iteration Method for Solving Fractional Riccati Differential Equation
International Differential Equations Volume 2010, Article ID 764738, 8 pages doi:10.1155/2010/764738 Research Article He s Variational Iteration Method for Solving Fractional Riccati Differential Equation
More informationComputers and Mathematics with Applications
Computers and Mathematics with Applications 1 (211) 233 2341 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Variational
More informationACTA UNIVERSITATIS APULENSIS No 20/2009 AN EFFECTIVE METHOD FOR SOLVING FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS. Wen-Hua Wang
ACTA UNIVERSITATIS APULENSIS No 2/29 AN EFFECTIVE METHOD FOR SOLVING FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS Wen-Hua Wang Abstract. In this paper, a modification of variational iteration method is applied
More informationNew Class of Boundary Value Problems
Inf. Sci. Lett. 1 No. 2, 67-76 (2012) Information Science Letters An International Journal 67 @ 2012 NSP Natural Sciences Publishing Cor. New Class of Boundary Value Problems Abdon Atangana Institute for
More informationThe variational homotopy perturbation method for solving the K(2,2)equations
International Journal of Applied Mathematical Research, 2 2) 213) 338-344 c Science Publishing Corporation wwwsciencepubcocom/indexphp/ijamr The variational homotopy perturbation method for solving the
More informationResearch Article Numerical Solution of the Inverse Problem of Determining an Unknown Source Term in a Heat Equation
Applied Mathematics Volume 22, Article ID 39876, 9 pages doi:.55/22/39876 Research Article Numerical Solution of the Inverse Problem of Determining an Unknown Source Term in a Heat Equation Xiuming Li
More informationComparison of Homotopy-Perturbation Method and variational iteration Method to the Estimation of Electric Potential in 2D Plate With Infinite Length
Australian Journal of Basic and Applied Sciences, 4(6): 173-181, 1 ISSN 1991-8178 Comparison of Homotopy-Perturbation Method and variational iteration Method to the Estimation of Electric Potential in
More informationdifferentiable functions in all arguments. Our aim is to minimize the quadratic objective functional (x T (t)qx(t)+u T (t)ru(t))dt, (2)
SOLVING NON-LINEAR QUADRATIC OPTIMAL... 49 differentiable functions in all arguments. Our aim is to minimize the quadratic objective functional J[x, u] = 1 2 tf t 0 (x T (t)qx(t)+u T (t)ru(t))dt, (2) subject
More informationHomotopy perturbation method for solving hyperbolic partial differential equations
Computers and Mathematics with Applications 56 2008) 453 458 wwwelseviercom/locate/camwa Homotopy perturbation method for solving hyperbolic partial differential equations J Biazar a,, H Ghazvini a,b a
More informationThe variational iteration method for solving linear and nonlinear ODEs and scientific models with variable coefficients
Cent. Eur. J. Eng. 4 24 64-7 DOI:.2478/s353-3-4-6 Central European Journal of Engineering The variational iteration method for solving linear and nonlinear ODEs and scientific models with variable coefficients
More informationConformable variational iteration method
NTMSCI 5, No. 1, 172-178 (217) 172 New Trends in Mathematical Sciences http://dx.doi.org/1.2852/ntmsci.217.135 Conformable variational iteration method Omer Acan 1,2 Omer Firat 3 Yildiray Keskin 1 Galip
More informationA New Technique of Initial Boundary Value Problems. Using Adomian Decomposition Method
International Mathematical Forum, Vol. 7, 2012, no. 17, 799 814 A New Technique of Initial Boundary Value Problems Using Adomian Decomposition Method Elaf Jaafar Ali Department of Mathematics, College
More informationNumerical comparison of methods for solving linear differential equations of fractional order
Chaos, Solitons and Fractals 31 (27) 1248 1255 www.elsevier.com/locate/chaos Numerical comparison of methods for solving linear differential equations of fractional order Shaher Momani a, *, Zaid Odibat
More informationVARIATIONAL ITERATION HOMOTOPY PERTURBATION METHOD FOR THE SOLUTION OF SEVENTH ORDER BOUNDARY VALUE PROBLEMS
VARIATIONAL ITERATION HOMOTOPY PERTURBATION METHOD FOR THE SOLUTION OF SEVENTH ORDER BOUNDARY VALUE PROBLEMS SHAHID S. SIDDIQI 1, MUZAMMAL IFTIKHAR 2 arxiv:131.2915v1 [math.na] 1 Oct 213 Abstract. The
More informationOn the Numerical Solutions of Heston Partial Differential Equation
Math Sci Lett 4, No 1, 63-68 (215) 63 Mathematical Sciences Letters An International Journal http://dxdoiorg/112785/msl/4113 On the Numerical Solutions of Heston Partial Differential Equation Jafar Biazar,
More informationSolution of the first order linear fuzzy differential equations by some reliable methods
Available online at www.ispacs.com/jfsva Volume 2012, Year 2012 Article ID jfsva-00126, 20 pages doi:10.5899/2012/jfsva-00126 Research Article Solution of the first order linear fuzzy differential equations
More informationThe Solution of the One Species Lotka-Volterra Equation Using Variational Iteration Method ABSTRACT INTRODUCTION
Malaysia Joural of Mahemaical Scieces 2(2): 55-6 (28) The Soluio of he Oe Species Loka-Volerra Equaio Usig Variaioal Ieraio Mehod B. Baiha, M.S.M. Noorai, I. Hashim School of Mahemaical Scieces, Uiversii
More informationHomotopy Perturbation Method for the Fisher s Equation and Its Generalized
ISSN 749-889 (print), 749-897 (online) International Journal of Nonlinear Science Vol.8(2009) No.4,pp.448-455 Homotopy Perturbation Method for the Fisher s Equation and Its Generalized M. Matinfar,M. Ghanbari
More informationResearch Article Series Solution of the Multispecies Lotka-Volterra Equations by Means of the Homotopy Analysis Method
Hindawi Publishing Corporation Differential Equations and Nonlinear Mechanics Volume 28, Article ID 816787, 14 pages doi:1.1155/28/816787 Research Article Series Solution of the Multispecies Lotka-Volterra
More informationAdomian Decomposition Method with Laguerre Polynomials for Solving Ordinary Differential Equation
J. Basic. Appl. Sci. Res., 2(12)12236-12241, 2012 2012, TextRoad Publication ISSN 2090-4304 Journal of Basic and Applied Scientific Research www.textroad.com Adomian Decomposition Method with Laguerre
More informationResearch Article On a New Reliable Algorithm
Hindawi Publishing Corporation International Journal of Differential Equations Volume 2009, Article ID 710250, 13 pages doi:10.1155/2009/710250 Research Article On a New Reliable Algorithm A. K. Alomari,
More informationInternational Journal of Modern Theoretical Physics, 2012, 1(1): International Journal of Modern Theoretical Physics
International Journal of Modern Theoretical Physics, 2012, 1(1): 13-22 International Journal of Modern Theoretical Physics Journal homepage:www.modernscientificpress.com/journals/ijmtp.aspx ISSN: 2169-7426
More informationALGORITHMS FOR NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS: A SELECTION OF NUMERICAL METHODS. Shaher Momani Zaid Odibat Ishak Hashim
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 31, 2008, 211 226 ALGORITHMS FOR NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS: A SELECTION OF NUMERICAL METHODS
More informationComparison of homotopy analysis method and homotopy perturbation method through an evolution equation
Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation Songxin Liang, David J. Jeffrey Department of Applied Mathematics, University of Western Ontario, London,
More informationApplication of Homotopy Perturbation Method in Nonlinear Heat Diffusion-Convection-Reaction
0 The Open Mechanics Journal, 007,, 0-5 Application of Homotopy Perturbation Method in Nonlinear Heat Diffusion-Convection-Reaction Equations N. Tolou, D.D. Ganji*, M.J. Hosseini and Z.Z. Ganji Department
More informationExact Solutions of Fractional-Order Biological Population Model
Commun. Theor. Phys. (Beijing China) 5 (009) pp. 99 996 c Chinese Physical Society and IOP Publishing Ltd Vol. 5 No. 6 December 15 009 Exact Solutions of Fractional-Order Biological Population Model A.M.A.
More informationThe Modified Adomian Decomposition Method for. Solving Nonlinear Coupled Burger s Equations
Nonlinear Analysis and Differential Equations, Vol. 3, 015, no. 3, 111-1 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/nade.015.416 The Modified Adomian Decomposition Method for Solving Nonlinear
More informationRELIABLE TREATMENT FOR SOLVING BOUNDARY VALUE PROBLEMS OF PANTOGRAPH DELAY DIFFERENTIAL EQUATION
(c) 216 217 Rom. Rep. Phys. (for accepted papers only) RELIABLE TREATMENT FOR SOLVING BOUNDARY VALUE PROBLEMS OF PANTOGRAPH DELAY DIFFERENTIAL EQUATION ABDUL-MAJID WAZWAZ 1,a, MUHAMMAD ASIF ZAHOOR RAJA
More informationImproving the Accuracy of the Adomian Decomposition Method for Solving Nonlinear Equations
Applied Mathematical Sciences, Vol. 6, 2012, no. 10, 487-497 Improving the Accuracy of the Adomian Decomposition Method for Solving Nonlinear Equations A. R. Vahidi a and B. Jalalvand b (a) Department
More informationSolution for Partial Differential Equations Involving Logarithmic Nonlinearities
Australian Journal of Basic and Applied Sciences, 5(4): 60-66, 2011 ISSN 1991-8178 Solution for Partial Differential Equations Involving Logarithmic Nonlinearities Majid Amirfakhrian and Somayeh Keighobadi
More informationExact Solutions for Systems of Volterra Integral Equations of the First Kind by Homotopy Perturbation Method
Applied Mathematical Sciences, Vol. 2, 28, no. 54, 2691-2697 Eact Solutions for Systems of Volterra Integral Equations of the First Kind by Homotopy Perturbation Method J. Biazar 1, M. Eslami and H. Ghazvini
More informationVariational Homotopy Perturbation Method for the Fisher s Equation
ISSN 749-3889 (print), 749-3897 (online) International Journal of Nonlinear Science Vol.9() No.3,pp.374-378 Variational Homotopy Perturbation Method for the Fisher s Equation M. Matinfar, Z. Raeisi, M.
More informationON NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS BY THE DECOMPOSITION METHOD. Mustafa Inc
153 Kragujevac J. Math. 26 (2004) 153 164. ON NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS BY THE DECOMPOSITION METHOD Mustafa Inc Department of Mathematics, Firat University, 23119 Elazig Turkiye
More informationHOMOTOPY PERTURBATION METHOD TO FRACTIONAL BIOLOGICAL POPULATION EQUATION. 1. Introduction
Fractional Differential Calculus Volume 1, Number 1 (211), 117 124 HOMOTOPY PERTURBATION METHOD TO FRACTIONAL BIOLOGICAL POPULATION EQUATION YANQIN LIU, ZHAOLI LI AND YUEYUN ZHANG Abstract In this paper,
More informationSolving nonlinear fractional differential equation using a multi-step Laplace Adomian decomposition method
Annals of the University of Craiova, Mathematics and Computer Science Series Volume 39(2), 2012, Pages 200 210 ISSN: 1223-6934 Solving nonlinear fractional differential equation using a multi-step Laplace
More informationSolution of the Coupled Klein-Gordon Schrödinger Equation Using the Modified Decomposition Method
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.4(2007) No.3,pp.227-234 Solution of the Coupled Klein-Gordon Schrödinger Equation Using the Modified Decomposition
More informationA Modified Adomian Decomposition Method for Solving Higher-Order Singular Boundary Value Problems
A Modified Adomian Decomposition Method for Solving Higher-Order Singular Boundary Value Problems Weonbae Kim a and Changbum Chun b a Department of Mathematics, Daejin University, Pocheon, Gyeonggi-do
More informationSafa Bozkurt Coşkun and Mehmet Tarik Atay. Received 13 December 2006; Revised 11 April 2007; Accepted 22 September 2007 Recommended by Josef Malek
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 27, Article ID 4272, 5 pages doi:.55/27/4272 Research Article Analysis of Convective Straight and Radial Fins with Temperature-Dependent
More informationCommun Nonlinear Sci Numer Simulat
Commun Nonlinear Sci Numer Simulat xxx (9) xxx xxx Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns Short communication Simple
More informationSOLUTION OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS BY ADOMIAN DECOMPOSITION METHOD
SOLUTION OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS BY ADOMIAN DECOMPOSITION METHOD R. C. Mittal 1 and Ruchi Nigam 2 1 Department of Mathematics, I.I.T. Roorkee, Roorkee, India-247667. Email: rcmmmfma@iitr.ernet.in
More informationThe Homotopy Perturbation Method for Solving the Modified Korteweg-de Vries Equation
The Homotopy Perturbation Method for Solving the Modified Korteweg-de Vries Equation Ahmet Yildirim Department of Mathematics, Science Faculty, Ege University, 351 Bornova-İzmir, Turkey Reprint requests
More informationHOMOTOPY ANALYSIS METHOD FOR SOLVING KDV EQUATIONS
Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (21), 89 98 HOMOTOPY ANALYSIS METHOD FOR SOLVING KDV EQUATIONS Hossein Jafari and M. A. Firoozjaee Abstract.
More informationNEW ANALYTICAL SOLUTION FOR NATURAL CONVECTION OF DARCIAN FLUID IN POROUS MEDIA PRESCRIBED SURFACE HEAT FLUX
THERMAL SCIENCE, Year 11, Vol. 15, Suppl., pp. S1-S7 1 Introduction NEW ANALYTICAL SOLUTION FOR NATURAL CONVECTION OF DARCIAN FLUID IN POROUS MEDIA PRESCRIBED SURFACE HEAT FLUX by Davood Domairy GANJI
More informationAn Alternative Approach to Differential-Difference Equations Using the Variational Iteration Method
An Alternative Approach to Differential-Difference Equations Using the Variational Iteration Method Naeem Faraz a, Yasir Khan a, and Francis Austin b a Modern Textile Institute, Donghua University, 1882
More informationOn Solutions of the Nonlinear Oscillators by Modified Homotopy Perturbation Method
Math. Sci. Lett. 3, No. 3, 229-236 (214) 229 Mathematical Sciences Letters An International Journal http://dx.doi.org/1.12785/msl/3315 On Solutions of the Nonlinear Oscillators by Modified Homotopy Perturbation
More informationResearch Article On the Numerical Solution of Differential-Algebraic Equations with Hessenberg Index-3
Discrete Dynamics in Nature and Society Volume, Article ID 474, pages doi:.55//474 Research Article On the Numerical Solution of Differential-Algebraic Equations with Hessenberg Inde- Melike Karta and
More informationA Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning
Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 22, 1097-1106 A Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning M. T. Darvishi a,, S.
More informationA Maple program for computing Adomian polynomials
International Mathematical Forum, 1, 2006, no. 39, 1919-1924 A Maple program for computing Adomian polynomials Jafar Biazar 1 and Masumeh Pourabd Department of Mathematics, Faculty of Science Guilan University
More informationCURRICULUM VITAE. Ahmad Sami Bataineh. October 16, 2014
CURRICULUM VITAE Ahmad Sami Bataineh October 16, 2014 1 PERSONAL DATA Name : AHMAD SAMI BATAINEH Date & Place of Birth : 10 October 1981, Irbid, Jordan Sex : Male Marital Status : Married Nationality :
More informationHomotopy Perturbation Method for Solving Systems of Nonlinear Coupled Equations
Applied Mathematical Sciences, Vol 6, 2012, no 96, 4787-4800 Homotopy Perturbation Method for Solving Systems of Nonlinear Coupled Equations A A Hemeda Department of Mathematics, Faculty of Science Tanta
More informationOn The Exact Solution of Newell-Whitehead-Segel Equation Using the Homotopy Perturbation Method
On The Exact Solution of Newell-Whitehead-Segel Equation Using the Homotopy Perturbation Method S. Salman Nourazar, Mohsen Soori, Akbar Nazari-Golshan To cite this version: S. Salman Nourazar, Mohsen Soori,
More informationHybrid Functions Approach for the Fractional Riccati Differential Equation
Filomat 30:9 (2016), 2453 2463 DOI 10.2298/FIL1609453M Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Hybrid Functions Approach
More informationExact Analytic Solutions for Nonlinear Diffusion Equations via Generalized Residual Power Series Method
International Journal of Mathematics and Computer Science, 14019), no. 1, 69 78 M CS Exact Analytic Solutions for Nonlinear Diffusion Equations via Generalized Residual Power Series Method Emad Az-Zo bi
More informationThe Use of Sumudu Decomposition Method for Solving Predator-Prey Systems
Math. Sci. Lett. 5, No., 285-289 (2016) 285 Mathematical Sciences Letters An International Journal http://dx.doi.org/10.18576/msl/05010 The Use of Sumudu Decomposition Method for Solving Predator-Prey
More informationSoliton solution of the Kadomtse-Petviashvili equation by homotopy perturbation method
ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 5 (2009) No. 1, pp. 38-44 Soliton solution of the Kadomtse-Petviashvili equation by homotopy perturbation method H. Mirgolbabaei
More informationSolving Zhou Chaotic System Using Fourth-Order Runge-Kutta Method
World Applied Sciences Journal 21 (6): 939-944, 2013 ISSN 11-4952 IDOSI Publications, 2013 DOI: 10.529/idosi.wasj.2013.21.6.2915 Solving Zhou Chaotic System Using Fourth-Order Runge-Kutta Method 1 1 3
More informationAn assessment of a semi analytical AG method for solving two-dimension nonlinear viscous flow
Int. J. Nonlinear Anal. Appl. 6 015 No., 47-64 ISSN: 008-68 electronic http://dx.doi.org/10.075/ijnaa.015.70 An assessment of a semi analytical AG method for solving two-dimension nonlinear viscous flow
More informationResearch Article Solutions of the Force-Free Duffing-van der Pol Oscillator Equation
International Differential Equations Volume 211, Article ID 852919, 9 pages doi:1.1155/211/852919 Research Article Solutions of the Force-Free Duffing-van der Pol Oscillator Equation Najeeb Alam Khan,
More informationSolution of Differential Equations of Lane-Emden Type by Combining Integral Transform and Variational Iteration Method
Nonlinear Analysis and Differential Equations, Vol. 4, 2016, no. 3, 143-150 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/nade.2016.613 Solution of Differential Equations of Lane-Emden Type by
More informationResearch Article Optimal Parametric Iteration Method for Solving Multispecies Lotka-Volterra Equations
Discrete Dynamics in Nature and Society Volume 2012, Article ID 842121, 10 pages doi:10.1155/2012/842121 Research Article Optimal Parametric Iteration Method for Solving Multispecies Lotka-Volterra Equations
More informationEXACT TRAVELING WAVE SOLUTIONS FOR NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USING THE IMPROVED (G /G) EXPANSION METHOD
Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 EXACT TRAVELIN WAVE SOLUTIONS FOR NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USIN THE IMPROVED ( /) EXPANSION METHOD Elsayed M.
More informationA Study On Linear and Non linear Schrodinger Equations by Reduced Differential Transform Method
Malaya J. Mat. 4(1)(2016) 59-64 A Study On Linear and Non linear Schrodinger Equations by Reduced Differential Transform Method T.R. Ramesh Rao a, a Department of Mathematics and Actuarial Science, B.S.
More informationHOMOTOPY PERTURBATION METHOD FOR SOLVING THE FRACTIONAL FISHER S EQUATION. 1. Introduction
International Journal of Analysis and Applications ISSN 229-8639 Volume 0, Number (206), 9-6 http://www.etamaths.com HOMOTOPY PERTURBATION METHOD FOR SOLVING THE FRACTIONAL FISHER S EQUATION MOUNTASSIR
More informationSolutions of the coupled system of Burgers equations and coupled Klein-Gordon equation by RDT Method
International Journal of Advances in Applied Mathematics and Mechanics Volume 1, Issue 2 : (2013) pp. 133-145 IJAAMM Available online at www.ijaamm.com ISSN: 2347-2529 Solutions of the coupled system of
More informationOn the Homotopy Perturbation Method and the Adomian Decomposition Method for Solving Abel Integral Equations of the Second Kind
Applied Mathematical Sciences, Vol. 5, 211, no. 16, 799-84 On the Homotopy Perturbation Method and the Adomian Decomposition Method for Solving Abel Integral Equations of the Second Kind A. R. Vahidi Department
More informationComputers and Mathematics with Applications. Adaptive anti-synchronization of chaotic systems with fully unknown parameters
Computers and Mathematics with Applications 59 (21) 3234 3244 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Adaptive
More informationNew Iterative Method for Time-Fractional Schrödinger Equations
ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 9 2013) No. 2, pp. 89-95 New Iterative Method for Time-Fractional Schrödinger Equations Ambreen Bibi 1, Abid Kamran 2, Umer Hayat
More informationAnalytical accuracy of the one dimensional heat transfer in geometry with logarithmic various surfaces
Cent. Eur. J. Eng. 4(4) 014 341-351 DOI: 10.478/s13531-013-0176-8 Central European Journal of Engineering Analytical accuracy of the one imensional heat transfer in geometry with logarithmic various surfaces
More informationThe Homotopy Perturbation Method (HPM) for Nonlinear Parabolic Equation with Nonlocal Boundary Conditions
Applied Mathematical Sciences, Vol. 5, 211, no. 3, 113-123 The Homotopy Perturbation Method (HPM) for Nonlinear Parabolic Equation with Nonlocal Boundary Conditions M. Ghoreishi School of Mathematical
More informationTHE DIFFERENTIAL TRANSFORMATION METHOD AND PADE APPROXIMANT FOR A FORM OF BLASIUS EQUATION. Haldun Alpaslan Peker, Onur Karaoğlu and Galip Oturanç
Mathematical and Computational Applications, Vol. 16, No., pp. 507-513, 011. Association for Scientific Research THE DIFFERENTIAL TRANSFORMATION METHOD AND PADE APPROXIMANT FOR A FORM OF BLASIUS EQUATION
More informationDIfferential equations of fractional order have been the
Multistage Telescoping Decomposition Method for Solving Fractional Differential Equations Abdelkader Bouhassoun Abstract The application of telescoping decomposition method, developed for ordinary differential
More informationVariation of Parameters Method for Solving Fifth-Order. Boundary Value Problems
Applied Mathematics & Information Sciences 2(2) (28), 135 141 An International Journal c 28 Dixie W Publishing Corporation, U. S. A. Variation of Parameters Method for Solving Fifth-Order Boundary Value
More informationAdomian s Decomposition Method for Solving Singular System of Transistor Circuits
Applied Mathematical Sciences, Vol. 6, 2012, no. 37, 1819-1826 Adomian s Decomposition Method for Solving Singular System of Transistor Circuits K. Krishnaveni veni.fairy@gmail.com S. Raja Balachandar
More informationApplication of the Differential Transform Method for the Nonlinear Differential Equations
American Journal of Applied Mathematics 27; 5(): 4- http://www.sciencepublishinggroup.com//aam doi:.64/.aam.275.2 ISSN: 233-43 (Print); ISSN: 233-6X (Online) Application of the Differential Transform Method
More informationA NEW SOLUTION OF SIR MODEL BY USING THE DIFFERENTIAL FRACTIONAL TRANSFORMATION METHOD
April, 4. Vol. 4, No. - 4 EAAS & ARF. All rights reserved ISSN35-869 A NEW SOLUTION OF SIR MODEL BY USING THE DIFFERENTIAL FRACTIONAL TRANSFORMATION METHOD Ahmed A. M. Hassan, S. H. Hoda Ibrahim, Amr M.
More informationNumerical Solution of 12 th Order Boundary Value Problems by Using Homotopy Perturbation Method
ohamed I. A. Othman, A.. S. ahdy and R.. Farouk / TJCS Vol. No. () 4-7 The Journal of athematics and Computer Science Available online at http://www.tjcs.com Journal of athematics and Computer Science
More informationSolving Fisher s Equation by Using Modified Variational Iteration Method
American Journal of Engineering, Technology and ociety 2017; 4(5): 74-78 http://www.openscienceonline.com/journal/ajets IN: 2381-6171 (Print); IN: 2381-618X (Online) olving Fisher s Equation by Using Modified
More informationThe Application of the Poincart-Transform to the Lotka-Volterra Model
J. Math. Biology 6, 67-73 (1978) Journal of by Springer-Verlag 1978 The Application of the Poincart-Transform to the Lotka-Volterra Model S. B. Hsu Department of Mathematics, University of Utah, Salt Lake
More informationCollege, Nashik-Road, Dist. - Nashik (MS), India,
Approximate Solution of Space Fractional Partial Differential Equations and Its Applications [1] Kalyanrao Takale, [2] Manisha Datar, [3] Sharvari Kulkarni [1] Department of Mathematics, Gokhale Education
More informationA note on the Adomian decomposition method for generalized Burgers-Fisher equation
INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATION Volume 11, 017 A note on the Adomian decomposition method for generalized Burgers-Fisher equation M. Meštrović, E. Ocvirk and D. Kunštek
More informationShiraz University of Technology. From the SelectedWorks of Habibolla Latifizadeh. Habibolla Latifizadeh, Shiraz University of Technology
Shiraz University of Technology From the SelectedWorks of Habibolla Latifizadeh 013 Variational iteration method for Nonlinear Oscillators: A comment on Application of Laplace Iteration method to Study
More informationVARIATION OF PARAMETERS METHOD FOR SOLVING SIXTH-ORDER BOUNDARY VALUE PROBLEMS
Commun. Korean Math. Soc. 24 (29), No. 4, pp. 65 615 DOI 1.4134/CKMS.29.24.4.65 VARIATION OF PARAMETERS METHOD FOR SOLVING SIXTH-ORDER BOUNDARY VALUE PROBLEMS Syed Tauseef Mohyud-Din, Muhammad Aslam Noor,
More informationA remark on a variable-coefficient Bernoulli equation based on auxiliary -equation method for nonlinear physical systems
A remark on a variable-coefficient Bernoulli equation based on auxiliary -equation method for nonlinear physical systems Zehra Pınar a Turgut Öziş b a Namık Kemal University, Faculty of Arts and Science,
More informationThe Chebyshev Collection Method for Solving Fractional Order Klein-Gordon Equation
The Chebyshev Collection Method for Solving Fractional Order Klein-Gordon Equation M. M. KHADER Faculty of Science, Benha University Department of Mathematics Benha EGYPT mohamedmbd@yahoo.com N. H. SWETLAM
More informationANALYSIS OF NONLINEAR DYNAMIC BEHAVIOUR OF NANOBEAM RESTING ON WINKLER AND PASTERNAK FOUNDATIONS USING VARIATIONAL ITERATION METHOD
ANALYSIS OF NONLINEAR DYNAMIC BEHAVIOUR OF NANOBEAM RESTING ON WINKLER AND PASTERNAK FOUNDATIONS USING VARIATIONAL ITERATION METHOD M. G. Sobamowo * and G. A. Oguntala Department of Mechanical Engineering,
More informationEXP-FUNCTION METHOD FOR SOLVING HIGHER-ORDER BOUNDARY VALUE PROBLEMS
Bulletin of the Institute of Mathematics Academia Sinica (New Series) Vol. 4 (2009), No. 2, pp. 219-234 EXP-FUNCTION METHOD FOR SOLVING HIGHER-ORDER BOUNDARY VALUE PROBLEMS BY SYED TAUSEEF MOHYUD-DIN,
More information