Literature Review: 1. Gamet, L. and Ducros (1999) 2. Abarbanel, S. and Ditkowski, A. (2000) 3. Mickens, R. (2001)

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1 Literature Review: 1. Gamet, L. and Ducros (1999) in their research paper they studied development of a fourth order compact scheme for approximation of first derivatives on non uniform grids. They present numerical analysis of truncation error. Convection equation for first derivative and diffusion equation for second derivative is considered. The ability of nonuniform mesh generalization of compact schemes is demonstrated to reproduce result. 2. Abarbanel, S. and Ditkowski, A. (2000) in their research paper, temporal behavior and rate of convergence of error bounds of finite difference approximations to partial differential equations is studied. They determined dependence of the error bounds on mesh size and time.for this purpose hyperbolic and parabolic partial differential equations are used. 3. Mickens, R. (2001) this paper is an introduction to non standard finite difference methods, which are useful to construct differential equations. In his paper, he described exact finite difference scheme, also rules for constructing non standard scheme with its application. 4. Fukagata,K. and Kesagi, N. (2002) they developed highly energy conservative finite difference method for cylindrical system. They proved that when approximate interpolation schemes are used then energy conservation in discretized space is satisfied. This holds for both equally and unequally spaced mesh on cylindrical coordinate system but not on Cartesian coordinates. 5. Farjadpur, A. and Roundy (2006)finite difference time domain method suffer from reduced accuracy due to discretization, for modeling discontinuous dielectric materials. They show that accuracy can be improved by using sub pixel smoothing, if it is properly designed. Also this scheme attains quadratic convergence. 6. Zhong,Q. and Zhi,L.(2006) In their research paper, They proposed numerical methods for solving non-linear Poisson-Boltzmann equation ψ = sinh ψ, where ψ is the electrostatic potential. A monotone iterative method was given for semi-linear partial differential equation of elliptic type. The modified central finite difference scheme is introduce. Numerical solutions agree with solutions obtained by adaptive finite element method.

2 7. Thankane, K.S. and Stys, T. (2009) in their research article, they present effective algorithms based on finite difference method for linear and non linear beam equations. Also they give the analysis of convergence of the algorithms. Solution of number of beam equations is given by designing Mathmatica Module. 8. McGee, S. and Seshaiyer, P. (2009) in their research paper application of finite difference methods for coupled flow interaction transport models are given. They considered a coupled two dimensional model with transient Navier-Stokes equation to model the blood flow in the vessel and Darcy s flow to model the plasma flow through the vessel wall. The advection diffusion equation is coupled with the velocities from the flows in the vessel and wall. The coupled chemical transport equations are discretized by the finite difference method and solved by using additive Schwartz method. 9. Dolicanin, C.B. and Nicolic, V.B. (2010) in their research paper, finite difference method is used to study of phenomenon in the theory of thin plates. FDM based on replacing differential equation into difference equation. This method can efficiently solve the problem of bending of thin plates. It is used to find solutions for the plate deflection, moments, stress, strain etc. 10. Islam, M.R. and Alias, N. (2010) Finite difference method is used to discretise a parabolic partial differential equation. They presented a mathematical simulation model using one dimensional parabolic equation. This model is regarding to moisture and temperature behavior of tropical herbs during dehyadration.here Jacobi, Gauss seidal and Red black Gauss seidal iterative methods are studied. It has proved that dehydration model is capable to simulate mass and temperature distribution through numerical methods approach. This mathematical simulation is time consuming and capable to reduce the risk of real experiments in actual process. 11. Mehra, M. and Patel, N. (2010) in their research paper three methods finite difference, spectral and Wavelet Galerkin Method (WGM) are compared. These methods are tested on Advection and Klein-Gordan equation.wgm gives better accuracy in comparison with other two methods. 12. Chambolle, A. and Levine, S.E. (2011In their research paper finite difference approximations to the variation problem is studied. They give dual formulation for an

3 upwind finite-difference method. They demonstrates numerically that the multiscale method is effective.they provide numerical examples for illustration qualitative and quantitative behavior of the solutions of numerical formulations. 13. Bothayana, S. H. (2011) in his research article used Adomian Decomposition Method for solving generalized Korteweg de Vries equations with boundary conditions. The solution is calculated in the series form. Adomian polynomials of the obtained series solution have been evaluated by mathematica program. 14. Mariam, A. (2011) in his research paper Adomian Decomposition Method is used for solving Goursat s problems for linear and non-linear hyperbolic equations of second order. Also the method used to solve system of non-linear hyperbolic equations and fourth order linear hyperbolic equations in which the attached conditions are given. Some examples having closed form solutions are also studied in detailed and results obtained indicate this approach is efficient and practical. 15. Wong, Y. and Guangrui, L. (2011)In their research paper Novel finite difference schemes are presented for solving the Helmholtz equation U xx +k 2 U=0, x (0,1) ; U(0)=1, U'(1) = ik U(1).The finite difference scheme is constructed so that solution of the discretized equations satisfies the solution of the Helmholtz equation exactly in the interior grid points and boundary points. Solution of Helmholtz equation is compared with central finite difference. Here solution by novel finite difference is accurately and efficiently computed. 16. Anderson, D. (2012) in his paper, He developed a new finite difference method for the computation of parameter sensitivities that is applicable to wide class of continuous time Markov chain. He constructed by coupling the perturbed and nominal processes naturally. He concludes that proposed method produces an estimator with lower variance than other methods. 17. Sungu, I.C. and Demir, H. (2012) in their research, hybrid method is introduced for non linear partial differential equation. The method is hybrid because different numerical methods, differential transform and finite differences are used in different subdomains.aim of this methods is to combine flexibility of differential transform and efficiency of finite difference. Differential transform is used for time operator and a finite

4 difference for discretization.hybrid approach is faster than corresponding finite difference method. The hybrid method provides iterative procedure to calculate accurate numerical solution. 18. Ibijola, E.A. and Obayomi, A.A. (2012) they derived non standard finite difference scheme for non autonomous ordinary differential equations. Non local approximation and renormalization of denominator function is given. Some numerical examples are given for construction of new non standard finite scheme. Scheme is compared with analytic solution and exact finite method. Also results are presented by graphs which show scheme is more reliable. 19. Zhang, J., Geng, X., et.al. (2012) they analyze two approaches for enhancing the accuracy of the standard second order finite difference scheme in solving one dimensional elliptic partial differential equations. These two approaches are the fourth order compact difference scheme and Richardson extrapolation for the fourth order accuracy. They studied the truncation error of these two approaches. They provide both analytic and numerical evidence to clarify difference between two approaches. 20. Morsy, S.A. and Azab, M.S. (2012) in their research paper new finite difference method is introduced which is known as logarithmic finite difference method (log FDM).This method is improved for solving linear or non linear higher order partial differential equations. They solve Kortweg de Vries Burger equation (KdVB).comparison to explicit finite difference method and exponential finite difference method is done. The numerical result shows that solution by log FDM gives high accuracy closed to analytic solution and no need to more conditions. 21. Bhanduvula, S. (2012) three dimensional advection-diffusion problems on Dirichlet boundary condition is considered.spliting scheme was proposed for the solution which allows replacing three dimensional problems into one dimension. Nonstandard finite difference scheme was proposed here. Numerical experiments are given for showing stability and convergence of finite difference scheme. 22. Kumar, A. and Pankaj, R. (2013) in their paper combined the Adomian Decomposition Method with Laplace Transform method present a new approach to solve a non-linear coupled and non-coupled Schrödinger equation with initial conditions. Their observation

5 is that this method does not need perturbation theory, linearization or weak non-linearity assumptions to obtain analytical and numerical solutions. They applied this method successfully for solving non-linear coupled and non-coupled Schrödinger equations and this method gives a solution of the partial differential equations in closed form while the mesh point technique only provide the approximation at mesh point. 23. Obaid, H., Ouifki R., et.al. (2013) They design a numerical method known as Non standard Finite difference method (NSFDM).This method is use to solve model of HIV represented by a nonlinear system of ordinary differential equations. The model describes the dynamic of HIV epidemic by partition of human population into susceptible and infectious subpopulations. They will use the forward difference approximations for the first derivative. The proposed schemes are unconditionally stable. 24. Mousa, M.M. and Reda, M. (2013) in their research paper the Adomian Decomposition Method (ADM) and the method of lines (MOL) are discussed to obtain solitary solutions of the Korteweg-de Vries equation (KdV). The numerical solutions by MOL are compared with the analytical solutions of Adomian Decomposition Method. In order to check the accuracy of the considered methods they have compared the obtained results with the exact ones. The observation is that the MOL is more effective and convenient than the Adomian Decomposition Method for solving such type of equations. 25. Gao, J. and Zhang, Y. (2013) they developed a staggered-grid finite difference scheme with variable order accuracy for porous media. They use method on dispersion relation to determine the orders of accuracy. The variation of parameters gives validity of scheme. Proposed method can decrease computational cost without reducing accuracy. 26. Izadian, J.l., Ranjbar, N., et.al. (2013)Application of Generalized finite difference method for solving elliptic equation on irregular mesh are given. This method is used to 3-D Poisson s equation with Dirichlet boundary condition on irregular grids in a cuboid.by using Taylor series expansion and least squares, partial derivatives are approximated. Numerical results are given by using examples, which shows efficiency of method. 27. Kalyani, P. and Ramchandra, P.S. (2013) In their research article, solution of one dimensional Heat equations with initial and boundary conditions are given by using finite

6 difference method and other numerical methods. Also they found the solution of Heat equation as a polynomial of two variables by using double interpolation. 28. Lakshmi, R. and Muthuselvi, M. (2013) Basic techniques of finite difference scheme for linear boundary value problem is applied for solving ordinary differential equation and they found numerical solution.matlab coding is also developed.numerial result is compared with analytic solution that agreed with exact solution. 29. Desale, S.V. and Pradhan, V.H. (2013) in their research paper, boundary layer heat flow over a flat plate is discussed. Boundary layer equations are solved numerically by using implicit finite difference scheme known as Keller box method. The solutions obtained by this method are compared with Homotopy Perturbation Method which shows reliability and efficiency of the method. 30. Feng, K.C. and Lewis T. (2013) in their research paper, they developed a new framework for designing and analyzing convergent finite difference methods (FDM). This method approximating both classical and viscosity solutions of second order fully nonlinear partial differential equation (PDE). The aim of this paper is to extend first order fully non linear PDE into second order. They proposed classes of consistent and monotone FDM for second order fully nonlinear PDE.Numerical results presented for the performance of proposed FDM. 31. Ghods, A. and Mir, M. (2014) In this article evaluation of finite difference method in modeling a rectangular thin plate structure is given.in big construction systems subjected to arbitrary load, including complex boundary condition solution by analytic method is impossible.therfore differential equation are discretized by finite difference method.this method is relatively strong method for numerical solution of plate equation with different loading and support solutions conditions. 32. Pathak, A.K. and Doctor, H.D. (2014) they describe spline collocation, finite difference and finite element method for solving flow of electricity in a cable transmission lines. These methods are used to solve parabolic partial differential equation. Solution obtained from spline method is more accurate than finite difference and finite element method. Also spline implicit method has accurate results than spline explicit method.

7 33. Prasad, H.S. and Reddy, Y.N. (2014) In this research article fitted second order finite difference method is presented for solving singular perturbation two point boundary value problems. They have introduced fitting factor in finite difference method and obtained value from single perturbation. Absolute errors are presented to show efficiency of method. 34. Raffaele, D. and Beatrice, P.(2014)In their research paper they particularly aiming numerical solution of partial differential equation (PDE) with oscillatory solutions by considering diffusion equation u t = δu xx In this article, accurate and efficient solution of PDE by using finite difference method are focused. They considered a diffusion problem with mixed boundary conditions according to different finite differences. 35. Fukuchi, T. (2014) in this paper Finite difference method is applied on Cartesian coordinate systems oveomplex domain. In numerical calculations, all calculation on irregular domains reduce to regular domain.discretization in finite difference method is derived by using Taylors series expansion. Also systematic use of polynomial interpolation gives advantage in deriving higher order differences. Here Poisson equation is tested for numerical experiments. 36. George, K. (2015) two methods finite difference and finite element method (FEM) are studied computationally for solving two dimensional linear elliptic partial differential equations. These two methods have same accuracy with regular boundaries. But if boundaries are irregular then FEM gives better approximation. 37. Sharma, D. (2015) they give methodology for solving the advection-diffusion equation by using fuzzy finite difference scheme. Here explicit finite difference method is applied. They shows that fuzzy finite difference method is effective and accurate It also provides a tool for estimating uncertainty associated with concentration of radon, which is important to obtain uncertainty in the inhalation exposure by propagation through the uncertainty of breathing rate. 38. Riaz, R. and Ozair, A. (2015) they proposed an unconditionally stable Non standard finite difference scheme for solving non linear Riccati differential equation. Result obtained by this scheme is more accurate and efficient as compared to other techniques

8 such as Euler and RK-4 and differential transformation method. If we increase step size other scheme converges to false steady state. 39. Jater, S. (2015)In this paper method of lines is used.in this method elliptic and hyperbolic partial differential equation(pde) are converted into initial and boundary value problems in ordinary differential equation.this is done by replacing derivatives by using central difference method. The resulting ordinary differential equation is solved by using extended block Numerov type method. Proposed method may be applied to solve variety of elliptic and hyperbolic PDE, therefore this method is flexible. 40. Chaudhari, T.U. and Patel (2015) in this paper they proposed finite element method for numerical solution of nonlinear Poisson s equation with initial and bound conditions. A specific problem of Poisson s equation under certain assumption is discussed. It has been shown that increase in the number of grid points gives rise to an increase in accuracy of solution. Finite difference method will have difficulties due to domain with complex geometries; therefore finite element method becomes more evident. 41. Gulkac, V. (2015) an implicit finite difference method is developed here. This method is used for solving the Heat transfer equation with moving boundary problems. The efficiency of the method tested on two dimensional heat transfer equation and accuracy by Fourier series. Application of this method is easier as compare to other method such as spectral method, finite element method, heat balance integral method. 42. Panahil, and hassemi, (2016) in this research paper solution of Laplace s equation is given. Analytical solution is calculated by using separation of variables. Numerical solution are given by two methods finite difference method (FDM) and boundary element method (BEM).Both numerical methods are computed and compared with analytic solution. The results obtained by both the methods are agreed with analytic solution. 43. Mohamed, Salleh, et.al. (2016) in their research paper, the mathematical modeling of free convection boundary layer flow on solid sphere is considered. The partial differential equations are solved by using the Keller-box method, which is an implicit finite difference method in conjunction with Newton s method for linearization. This method is suitable for parabolic partial differential equation. Also this method is efficient to solve convective boundary layer problems involving partial differential equations.

9 After reviewing all this literature related to proposed research topic, it is observed that, Variational Iteration Method is not used for solving some specific non-linear partial differential equations. Therefore researcher used this method for solving such equations. The proposed method is capable of greatly reducing the size of calculation by maintaining high accuracy of numerical solution.