Open access peer-reviewed chapter

Study on Approximate Analytical Method with Its Application Arising in Fluid Flow

Written By

Twinkle R. Singh

Submitted: September 9th, 2020 Reviewed: April 2nd, 2021 Published: May 6th, 2021

DOI: 10.5772/intechopen.97548

Chapter metrics overview

348 Chapter Downloads

View Full Metrics


This chapter is about the, Variational iteration method (VIM); Adomian decomposition method and its modification has been applied to solve nonlinear partial differential equation of imbibition phenomenon in oil recovery process. The important condition of counter-current imbibition phenomenon as vi=−vn, has been considered here main aim, here is to determine the saturation of injected fluid Sixt during oil recovery process which is a function of distance ξ and time θ, therefore saturation Si is chosen as a dependent variable while xandt are chosen as independent variable. The solution of the phenomenon has been found by VIM, ADM and Laplace Adomian decomposition method (LADM). The effectiveness of our method is illustrated by different numerical.


  • Variational Iteration method (VIM)
  • Adomian decomposition method (ADM)
  • Laplace Adomian decomposition method (LADM)
  • nonlinear partial differential equations

1. Introduction

First, the variational iteration method was proposed by He [1] in 1998 and was successfully applied to autonomous ordinary differential equation, to nonlinear partial differential equations with variable coefficients. In recent times a good deal of attention has been devoted to the study of the method. The reliability of the method and the reduction in the size of the computational domain give this method a wide applicability. The VIM based on the use of restricted variations and correction functional which has found a wide application for the solution of nonlinear ordinary and partial differential equations, e.g., [2, 3, 4, 5, 6, 7, 8, 9, 10]. This method does not require the presence of small parameters in the differential equation, and provides the solution (or an approximation to it) as a sequence of iterates. The method does not require that the nonlinearities be differentiable with respect to the dependent variable and its derivatives and whereas the Adomian decomposition method was before the Nineteen Eighties, it was developed by Adomian [11, 12] for solving linear or nonlinear ordinary, partial and Delay differential equations. A large type of issues in mathematics, physics, engineering, biology, chemistry and other sciences have been solved using the ADM, as reported by many authors [13]. The Adomian decomposition method (ADM) [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] is well set systematic method for practical solution of linear or nonlinear and deterministic or stochastic operator equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), integral equations, integro-differential equations, etc. The ADM is considered as a powerful technique, which provides efficient algorithms for analytic approximate solutions and numeric simulations for real-world applications in the applied sciences and engineering. It allows us to solve both nonlinear initial value problems (IVPs) and boundary value problems (BVPs) [17, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46] without unphysical restrictive assumptions such as required by linearization, perturbation, ad hoc assumptions, guessing the initial term or a set of basic functions, and so forth. The accuracy of the analytic approximate solutions obtained can be verified by direct substitution. More advantages of the ADM over the variational iteration method is mentioned in Wazwaz [22, 28]. A key notion is the Adomian polynomials, which are tailored to the particular nonlinearity to solve nonlinear operator equations. A key concept of the Adomian decomposition series is that it is computationally advantageous rearrangement of the Banach-space analog of the Taylor expansion series about the initial solution component function, which permits solution by recursion. The selection behind choice of decomposition is nonunique, which provides a valuable advantage to the analyst, permitting the freedom to design modified recursion schemes for ease of computation in realistic systems.

Same way Laplace Adomian’s Decomposition Method (LADM) was first introduced by Khuri [47, 48]. The Laplace Adomian Decomposition Method (LADM) is formed with combination of the Adomian Decomposition Method (ADM) Adomian [29, 49] and Laplace transforms. LADM is a promising method and has been applied in solving various nonlinear systems of differential equations [36, 50, 51, 52, 53, 54, 55, 56]. In a variety of applied sciences, systems of partial differential equations have attracted much attention e.g. [50, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75]. The general ideas and the essentiality of these systems are of wide applicability. Agadjanov [56] solved Duffing equation with the help of LDM. Elgazery [51, 76] had applied Laplace decomposition method for the solution of Falkner-Skan equation.

In the solution procedure of VIM; many repeated computations and computations of the unneeded forms, which take more time and effort beyond it, so a modification has been shown to reduce these unneeded forms.

On the other hand, few researchers have been discussed imbibition phenomenon in homogenous porous media with different point of view for example, researchers taking different perspectives for this phenomenon; [77, 78] and some others have analyzed it for homogeneous porous medium.

In this Present investigated model, Imbibition takes place over a small part of a large oil formatted region taken as a cylindrical piece of homogeneous porous medium. In this model, we have considered the important condition of counter-current imbibition phenomenon as vi=vn, Our purpose is to determine the saturation of injected fluid Sixt during oil recovery process which is a function of distance ξ and time θ, therefore saturation Si has been chosen as a dependent variable while xandt are chosen as independent variable.


2. Imbibition phenomenon

It is the process by which a wetting fluid displaces a non-wetting fluid the initially saturates a porous sample, by capillary forces alone. Suppose a sample is completely saturated with a non-wetting fluid, and same wetting fluid is introduced on its surface. There will be spontaneous flow of wetting fluid into the medium, causing displacement of the non-wetting fluid. This is called imbibition phenomenon. The rate of imbibition is greater if the wettability of the porous medium, by the imbibed fluid, is higher.

The mathematical condition for imbibition phenomenon is given by Scheidegger [78]); viz,


Where vi&vn are the seepage velocities of injected & native liquids respectively.

The relation between relative permeability and phase-saturation,


Where ki&kn denotes fictitious relative permeability. Si&Sn denotes saturations of injected and native liquids respectively.


3. Mathematical structure of the model

According to the Darcy’s law, the basic equations of the phenomenon as; [78]


Where vi and vn are the seepage velocities, ki and kare the relative permeabilities δi and δn are the kinematic viscosities (which are constants), pi and pn are pressure of the injected and native liquid respectively, φandK are the porosity and the permeability of the homogeneous porous medium; Si is the saturation of the injected liquid; pc is the capillary pressure and t is the time. The co-ordinate x is measured along the axis of the cylindrical medium, the origin being located at the imbibition face x=0.

Combing equations (1)-(5) and using the relation for capillary pressure as,

pc=βSi [70], we get,


Where D(Si) = kiknδnki+δikn and β being small capillary pressure coefficient.

It is assumed is that an average value of D(Si) =D¯ (Si)

Using the transformation,


Eq. (7), becomes;




By the Hopf-Cole transformation [79, 80] equation (9) reduces to the Burger’s equation.


With the condition

Siξ0=Si0eξ at time θ = 0 and ξ > 0

3.1 Solution of the Burger’s equation by variational iteration method

To add the basic concepts of VIM, considering the below mentioned nonlinear partial differential equations:


Where L=t,R is a linear operator which has partial derivatives with respect to x, Nu(x,t) is a nonlinear term and g(x,t) is an inhomogeneous term.

As per the VIM [6, 7];


Where λ is called a general Lagrange multiplier [81, 82] which can be identified optimally via vatiational theory, RUn¯ and NUn¯ are considered as restricted variations,

i.e. δRUn¯=0,δNUn¯=0 calculating variation with respect to Un;


The Lagrange multiplier, therefore, can be considered as λ=-1.

Now, substituting the multiplier in (12), then


With the constrain

Siξ0=Si0eξ at time θ = 0 and ξ > 0

To solve equation (10) by VIM, substituting in equation (14) by


& g(x,t) = 0

And can obtain the following variational iteration formula:


Using (14), the approximate solutions Unxtare obtained by substituting;


Approximate solutions are given below;




And so on…..

Notes on VIM

From the analysis we can observed is this:

  1. VIM can contain a series solution not exactly like ADM.

  2. VIM needs many modifications to overcome the wasted time in the repeated calculations and unneeded terms.

To overcome these problems, following ADM and LADM is suggested.

Now applying ADM to equation (10); we get


And recursive relation is:




and so on…

Now, applying (LADM) Laplace transform with respect to t on both sides of (10);


And so on…


4. Interpretation

It is concluded that for the non linear partial differential equation of imbibitions phenomenon in oil recovery process, through graphs, it has been observed that the saturation of injected water during imbibition, increases and it is noted that LADM gives faster accuracy compare to VIM and ADM (Figures 17).

Figure 1.

Plot of Saturation Siξθ versus ξ for VIM Solution.

Figure 2.

Plot of Saturation Siξθ versus θ for VIM Solution.

Figure 3.

3-Dimensional VIM Solution.

Figure 4.

Plot of Saturation Siξθ versus ξ for ADM Solution.

Figure 5.

Plot of Saturation Siξθ versus ξ for LADM Solution.

Figure 6.

Plot of Saturation Siξθ versus θ for LADM Solution.

Figure 7.

3-Dimensional LADM Solution


5. Conclusions

The VIM, the ADM and the LADM are successfully applied to Burger’s equation. The results which are obtained by ADM are a powerful mathematical tool to solve nonlinear partial differential equation. It has been noted that this method is reliable and requires fewer computations; and scheme LADM gives better and very faster accuracy in comparison with VIM.



The authors are thankful to Applied Mathematics and Humanities Department of S. V. National Institute of Technology, Surat for the encouragement and facilities.


  1. 1. He, J.H. (1998). Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Engrg. 167, 69–73
  2. 2. Al-Hayani, W.(2011). Adomian decomposition method with Green’s function for sixth- Orderboundary value problems, Comp. Math. Appl. 61, 1567–1575
  3. 3. Al-Hayani, W.and. Casasús, L (2005). Approximate analytical solution of fourth order Boundaryvalue problems, Numer. Algorithms 40, 67–78
  4. 4. Abdou, M.A. and Soliman, A.A. (2005). New applications of variational iteration method, Physica D 211 (1-2), 1-8
  5. 5. Abdou, M.A. and Soliman, A.A. (2005). Variational iteration method for solving Burger’s and coupled Burger’s equations, J. Comput. Appl. Math. 181(20), 245-251
  6. 6. Momani, S.,Abuasad, S. (2005). Application of He’s varitional iteration method to Helmholtz equation, Chaos Solitons & Fractals 27, 1119-1123
  7. 7. Odibat, Z.M. , Momani,S.(2006). Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul. 7 (1), 27- 34
  8. 8. Soliman, A. A. (2006). Numerical simulation and explicit solutions of KdV–Burgers’ and Lax’s seventh-order KdV equations, Chaos Solitons & Fractals; 29 (2), 294-302
  9. 9. Scott, M. R. (1973). Invariant Imbedding and its Applications to Ordinary Differential Equations, Addison-vesley
  10. 10. Sweilam, N.Hand Khader, M.M. (2007). Variational iteration method for one dimensional nonlinear thermoelasticity, Chaos Solitons & Fractals, 32 (1),145-149
  11. 11. Adomian,G.91989). Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer Academic, Dordrecht
  12. 12. Adomian, G. (1994). Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht
  13. 13. Serrano, S.E. (2010). Hydrology for Engineers, Geologists, and Environmental rofessionals: An Integrated Treatment of Surface, Subsurface, and Contaminant Hydrology, Second Revised Edition, HydroScience, Ambler, PA
  14. 14. Adomian, G. (1983) R. Rach, Inversion of nonlinear stochastic operators, J. Math. Anal. Appl.91, 39–46
  15. 15. Adomian,G. (1983). Stochastic Systems, Academic, New York
  16. 16. Adomian, G.(1986). Nonlinear Stochastic Operator Equations, Academic, Orlando, FL
  17. 17. Adomian, G, Rach, R. (1993). A new algorithm for matching boundary conditions in decomposition solutions, Appl. Math. Comput. 58, 61–68
  18. 18. Bellman, R. E. and Adomian, G. (1985). Partial Differential Equations: New Methods for their Treatment and Solution, D.Reidel, Dordrecht
  19. 19. Bellomo, N.and Riganti, R. (1987). Nonlinear Stochastic System Analysis in Physics and Mechanics, World Scientific, Singapore and River Edge, NJ
  20. 20. Cherruault, Y. (1998). Modèles et méthodes mathématiques pour les sciences du vivant,Presses Universitaires de France, Paris
  21. 21. Wazwaz, A. M. (1997). A First Course in Integral Equations, World Scientific, Singapore and River Edge, NJ
  22. 22. Wazwaz, A. M. (2002). Partial Differential Equations: Methods and Applications, A. A. Balkema,Lisse, The Netherlands
  23. 23. Wazwaz, A. M. (2009). Partial Differential Equations and Solitary Waves Theory, Higher Education Press, Beijing, and Springer-Verlag, Berlin
  24. 24. Wazwaz, A.M. (2011). Linear and Nonlinear Integral Equations: Methods and Applications,Higher EducationPress, Beijing, and Springer-Verlag, Berlin
  25. 25. Wazwaz, A.M. (2000). Approximate solutions to boundary value problems of higher order by themodified decomposition method, Comput. Math. Appl. 40, 679–691
  26. 26. Wazwaz, A.M. (2000). The modified Adomian decomposition method for solving linear and nonlinear boundary value problems of tenth-order and twelfth-order, Int. J. Nonlinear Sci. Numer. Simul. 1, 17–24
  27. 27. Wazwaz, A.M. (2000). A note on using Adomian decomposition method for solving boundary value problems, Found. Phys. Lett. 13, 493–498
  28. 28. Wazwaz, A.M. (2000). The numerical solution of special eighth-order boundary value problems by the modified decomposition method, Neural Parallel Sci. Comput. 8,133–146
  29. 29. Adomian, G.(1994). Solving Frontier Problems of Physics: Decomposition method.Kluwer, Boston, MA
  30. 30. Al-Sawalha, M.M., Noorani, M.S.M. and. Hashim, I. (2008). Numerical experiments on the hyperchaotic Chen system by the Adomian decomposition method, Int. J. Comput. Methods 5, 403–412
  31. 31. Bigi, D. and Riganti, R. (1986). Solution of nonlinear boundary value problems by the decomposition method, Appl. Math. Modelling 10, 49–52
  32. 32. Casasús L.and Al-Hayani, W.(2000). The method of Adomian for a nonlinear boundary value problem, Rev. Acad. Canar. Cienc. 12, 97–105
  33. 33. Duan, J.S. and Rach,R. (2011). New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods, Appl. Math. Comput. 218, 2810–2828
  34. 34. Ebadi G. and Rashedi, S. (2010). The extended Adomian decomposition method for fourth order boundary value problems,Acta Univ. Apulensis 22, 65–78
  35. 35. He,J.H. (1997). A new approach to nonlinear partial differential equations, Comm. Nonlinear Sci. Numer. Simul. 2 (1997) 230–235
  36. 36. Hussain, M. and Khan, M. (2010). Modied Laplace decomposition method, Appl. Math. Sci, 36, 1769-1783
  37. 37. Jun-Sheng Duan a, Randolph Rach , Dumitru B˘aleanu , Abdul-Majid Wazwaz. (2012). A review of the Adomian decomposition method and its applications to fractional differential equations. Commun. Frac. Calc. 3 (2), 73 – 99
  38. 38. Jasem Fadaei. (2011). Application of Laplace − Adomian Decomposition Method on Linear and Nonlinear System of PDEs. Applied Mathematical Sciences, Vol. 5, 27, 1307 –1315
  39. 39. Jang, B. (2008). Two-point boundary value problems by the extended Adomian decomposition method, J. Comput. Appl.Math. 219, 253–262
  40. 40. Khan, Y. (2009). An effective modification of the Laplace decomposition method for Nonlinear equations, International Journal of Nonlinear Sciences and Numerical Simulation, 10, 1373-1376
  41. 41. Lesnic, D. (2008). The decomposition method for nonlinear, second-order parabolic partial differential equations, Int. J. Comput. Math. Numeric. Simulat. 1, 207–233
  42. 42. Rach R.and. Baghdasarian, A . (1990). On approximate solution of a nonlinear differential equation, Appl. Math. Lett. 3, 101–102
  43. 43. Tatari, M.and Dehghan, M.(2006). The use of the Adomian decomposition method for solving multipoint boundary value problems, Physica Scripta 73, 672–676
  44. 44. Tsai, P.Y. and Chen, C.K. (2010). An approximate analytic solution of the nonlinear Riccati differential equation, J. Frank. Inst. 347, 1850–1862
  45. 45. Wazwaz A.M. and Rach, R. (2011). Comparison of the Adomian decomposition methodand the variational iteration method for solving the Lane-Emden equations of the first and second kinds, Kybernetes 40, 1305–1318
  46. 46. Wazwaz, A.M. (2012). A reliable study for extensions of the Bratu problem with boundary conditions, Math. Methods Appl. Sci. 35 (2012) 845–856
  47. 47. Khuri, S. A. (2001). A Laplace decomposition algorithm applied to a class ofNonlineardifferential equations,” J. Math. Annl. Appl., 4, 141-155
  48. 48. Khuri, S. A. (2004). A new approach to Bratus problem,” Appl. Math. Comp., 147 (2004) 31-136
  49. 49. Adomian, G.(1988). A review of the decomposition method in applied mathematics," Journal of mathematical analysis and applications, vol. 135, 501-544
  50. 50. Duan, J.S. and Rach, R. (2011). A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations, Appl.Math. Comput. 218, 4090–4118
  51. 51. Elgazery, S. Nasser. (2008). Numerical solution for the Falkner-Skan equation, Chaos, Solitons and Fractals, 35, 738 - 746
  52. 52. Fadaei, J. and Moghadam, M.M. (2012). Numerical Solution of Systems of Integral Differential equations by Using Modied Laplace Adomian Decomposition Method, World Applied Sciences Journal 19, 1818-1822
  53. 53. Ongun, M.Y. (2011). The Laplace Adomian Decomposition Method for solving a model for HIV infection of CD4+T cells, Math. Comp.Modell, 53, 597-603
  54. 54. Saei, F., Dastmalchi, F.,Misagh, D.,Zahiri, Y.,Mahmoudi, M., Salehian, V.and Rafati Maleki, N. (2013). New Application of Laplace Decomposition Algorithm For Quadrtic Riccati Differential Equation by Using Adomian’s Polynomials, Life Science Journal,10, 3s
  55. 55. Yusufoglu, E. (2006). Numerical solution of Dung equation by the Laplace decomposition algorithm, Appl. Math. Comput, 177, 572-580
  56. 56. Yusufoglu, Elcin,(Agadjanov), (2006). Numerical solution of Duffing equation by the Laplace decomposition algorithm, Appl. Math. Comput. 177, 572 - 580
  57. 57. Abbasbandy, S.(2006). Iterated He’s homotopy Perturbation method for quadratic Riccati differential equation, Applied Mathematics and Computation 175, 581-589
  58. 58. Adomian G and Rach, R. (1993). Analytic solution of nonlinear boundary-value problems in several dimensions by decomposition, J. Math. Anal. Appl. 174,118–137
  59. 59. Al-Mazmumy, M.and Al-Malki,H. (2015). The modified Adomian Decomposition Method for solving Nonlinear Coupled Burger’s Euations, Nonlinear Analysis and Differential Equations, Vol. 3, No.3, 111-122
  60. 60. Dehghan, M.and Tatari, M. (2010). Finding approximate solutions for a class of third-order non-linear boundary value problems via the decomposition method of Adomian, Int. J. Comput. Math. 87,1256–1263
  61. 61. Ebaid, A.E. (2010). Exact solutions for a class of nonlinear singular two-point boundary value problems: The decomposition method, Z. Naturforsch. 65a (2010) 1–6
  62. 62. Ebaid, A.E. (2011). A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method, J. Comput. Appl. Math. 235, 1914–1924
  63. 63. Geny, F.Z., Lin, I.Z., Cui, M.G.(2009). A piecewise variational iteration method for Riccati differential equations, Computers and mathematics with Applications 58, 2518-2522
  64. 64. Hashim, I., Adomian, G. (2006). Decomposition method for solving BVPs for fourth- order integro-differential equations, J. Comput. Appl. Math. 193, 658–664
  65. 65. He, J.H. (2004). Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos Solitons Fractals 19, 847–851
  66. 66. He, J.H. (2000). Variational iteration Method for autonomons ordinary differential systm, Applied Mathematics and Computation 114, 115-123
  67. 67. He, J.H.(1999). Variational iteration Method – a kind of non-linear analytical technique; some examples, International journal of Non-linear Mechanics 34, 699-708
  68. 68. He, J.H. (1998). Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg. 167, 57–68
  69. 69. Onur Kıymaz, Ayşegül Çetinkaya. (2010). Variational Iteration Method for a Class of Nonlinear Differential Equations. Int. J. Contemp. Math. Sciences, Vol. 5, 37:1819 – 1826
  70. 70. Oroveanu,T.(1963). Scurgerea fluiidelor prin medii poroase neomogene, Editura Academiei Republicii Populare Romine,92, 328
  71. 71. Rach, R and Adomian, G. (1990). Multiple decompositions for computational convenience, Appl.Math. Lett. 3, 97–99
  72. 72. Sambath, M. and Balachandran, K. (2016). Laplace Adomian decomposition method for solving a fish farm model. Nonauton. Dyn. Syst. 3,104–111
  73. 73. Serrano, S.E. (2011). Engineering Uncertainty and Risk Analysis: A Balanced Approach to Probability, Statistics, Stochastic Modeling, and Stochastic Differential Equations, second Revised Edition, HydroScience, Ambler, PA
  74. 74. Taiwo, O. and Odetunde, O. (2010). On the numerical approximation of delay differential equations by a decomposition method," Asian Journal of Mathematics and Statistics,vol. 3, pp. 237-243
  75. 75. Verma, A.P. (1970): Perturbation solution in imbibition in a cracked porous medium of small inclination, IASH, 13, 1, 45-51
  76. 76. Elgazery, S.N. (2008). Numerical solution for the Falkner-Skan equation, Chaos, Solitons and Fractals, 35,738-746
  77. 77. Graham, R.A. and Richardson, J.G. (1959): Theory and applications of imbibition phenomena in recovery of oil, Trans, Aime, 216
  78. 78. Scheidegger, A.E. (1960). The Physics of Flow through Porous Media, University of Toronto press
  79. 79. Cole, J. D. (1951): On a quasilinear parabolic equation occurring in aerodynamics. Q. Appl. Math. g,225-236
  80. 80. Hopf, E. (1950): The partial differential equation ut + uux = ε uxx comm. Pure Appl. Math. 3, pp – 201-230
  81. 81. Tamer A. Abassy, Magdy A. El-Tawil, H. Fl. Zoheiry, Towards a modified Variational iteration method, Journal of Computational and Applied mathematics 207 (2007) 137-147
  82. 82. Tamer A. Abassy, Improved Adomian decomposition method; Computers and Mathematics with Applications 59 (2010) 42-54

Written By

Twinkle R. Singh

Submitted: September 9th, 2020 Reviewed: April 2nd, 2021 Published: May 6th, 2021