the decomposition method applied to systems of fredholm integral equations of the second kind in this paper the adomian decomposition method is applied to solve systems of linear and nonlinear fredholm integral equations of the second kind convergence of the method is proved and some examples are presented to illustrate the method

adomian decomposition a tool for solving a system of fractional differential equations abstract adomian decomposition method has been employed to obtain solutions of a system of fractional differential equations convergence of the method has been discussed with some illustrative examples in particular for the initial value problem d 1 y 1 d n y n t a y 1 y n t y i 0 c i i 1 n where a a i j is a real square matrix the solution turns out to be y x e 1 n 1 x 1 a 1 x n a n y 0 where e 1 n 1 denotes multivariate mittag leffler function defined for matrix arguments and a i is the matrix having ith row as a i 1 a i n and all other entries are zero fractional oscillation and bagley torvik equations are solved as illustrative examples

on series solution for second order semilinear parabolic ibvps abstract this paper presents a method to obtain semi analytical solutions to second order semilinear ibvps where the standard adomian decomposition may fail to yield nontrivial solutions or fail to produce the correct partial solutions in contrast to the standard adomian decomposition method the proposed solution method is distinguished by simultaneous inversion of the linear differential operators using eigenfunctions expansion representations the proposed method is applied to several examples of initial boundary value problems a linear advection diffusion problem a burgers equation and the deterministic kardar parisi zhang kpz equation it is shown that the dependence of the series solution on the truncation order n the number of eigenfunctions m and the diffusion coefficient is rather complex

an analytic study of fisher s equation by using adomian decomposition method in this paper we develop a framework to obtain exact solutions to fisher s equation and to a nonlinear diffusion equation of the fisher type by employing adomian decomposition method the proposed scheme is supported by examining nonlinear diffusion equations of the fisher type

a further consideration of partial solutions in the decomposition method abstract reference 1 showed that in partial differential equation problems involving linear operator terms with respect to x y z t considerable work is saved by solving only one of the linear operator equations to obtain a so called partial solution the heat equation was shown as a simple example which appeared to be a counter example because one partial solution was identically zero the l x u operator equation led to a zero solution while the l t u operator equation led to the correct solution e t sin x this is a case of asymptotic equality as t rather than equality it arises when boundary conditions are special as discussed in 1 in cases where one operator annihilates the series in a finite number of terms one partial solution may not satisfy the given conditions this paper provides further insight into cases where such an improper solution can be obtained and the resolution of such cases by appropriate expansion of the initial term without making a priori assumptions about the solution

revisit on partial solutions in the adomian decomposition method solving heat and wave equations the adomian decomposition method is considered in application to heat and wave equations the so called partial solution technique is used it is shown that the fundamental equation of the method is well defined only for certain types of boundary conditions in cases involving inhomogeneous boundary conditions improper results may be obtained by former method this paper presents a further insight into partial solutions in the decomposition method and the resolution of such cases

a new approach to the heat equation an application of the decomposition method abstract the author s decomposition method adomian stochastic systems academic press new york 1983 nonlinear stochastic operator equations academic press new york in press bellman and adomian partial differential equations methods for their treatment and solution reidel dordrecht 1984 adomian j comput appl math 2 1984 225 230 shown elsewhere to provide computable accurate approximate convergent solutions to nonlinear stochastic operator equations is applied to the heat equation

the decomposition approach to inverse heat conduction adomian s decomposition approach is employed for solving some inverse boundary value problems in heat conduction further the mollification method is applied to deal with noisy input data and obtain a stable approximate solution

a high order discontinuous galerkin method for 2d incompressible flows in this paper we introduce a high order discontinuous galerkin method for two dimensional incompressible flow in the vorticity stream function formulation the momentum equation is treated explicitly utilizing the efficiency of the discontinuous galerkin method the stream function is obtained by a standard poisson solver using continuous finite elements there is a natural matching between these two finite element spaces since the normal component of the velocity field is continuous across element boundaries this allows for a correct upwinding gluing in the discontinuous galerkin framework while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability the method is efficient for inviscid or high reynolds number flows optimal error estimates are proved and verified by numerical experiments

a new method for solving singular initial value problems in the second order ordinary differential equations singular initial value problems linear and nonlinear homogeneous and nonhomogeneous are investigated by using adomian decomposition method the solutions are constructed in the form of a convergent series a new general formula is established the approach is illustrated with few examples

experimentation with two step adomian decomposition method to solve evolution models the two step adomian decomposition method tsadm is derived by separate the standard adomian decomposition method into two steps although the tsadm needs only a slight variation from the standard adomian decomposition method experimentation shows that the tsadm is also promising for systems of inhomogeneous differential and integral equations hyperbolic partial differential equations and singular initial value problems

a new algorithm for solving classical blasius equation in this paper a reliable algorithm is employed to investigate the classical blasius equation the algorithm is based mainly on applying the adomian decomposition method to the transformation of the blasius equation the results demonstrate reliability and efficiency of the proposed algorithm

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

analytic solution of nonlinear boundary value problems in several dimensions by decomposition abstract nonlinear partial differential equations are systematically solved by the decomposition method of g adomian for general boundary conditions described by boundary operator equations linear partial differential equations and linear or nonlinear ordinary differential equations become special cases all dealt with by a single procedure

a new approach to nonlinear partial differential equations abstract the author s decomposition method for the solution of operator equations which may be nonlinear and or stochastic is generalized to multidimensional cases

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

a new computational method for laplace transforms by decomposition method in this paper adomian decomposition method is applied to develop a new computational method for laplace transforms some examples are provided to illustrate the simplicity and reliability of the new method

the restrictions and improvement of the adomian decomposition method the adomian decomposition method is a creative and effective method for exactly solving functional equations of various kinds in this paper the principle of the decomposition method is described and its restrictions are discussed then an improvement is proposed which overcomes the restrictions and complements the ability of the adomian decomposition method some concrete examples are also studied to illustrate the validity of the improvement

finite difference methods for 3d viscous incompressible flows in the vorticity vector potential formulation on nonstaggered grids simple efficient and accurate finite difference methods are introduced for 3d unsteady viscous incompressible flows in the vorticity vector potential formulation on nonstaggered grids two different types of methods are discussed they differ in the implementation of the normal component of the vorticity boundary condition and consequently the enforcement of the divergence free condition for vorticity both second order and fourth order accurate schemes are developed a detailed accuracy test is performed revealing the structure of the error and the effect of how the convective terms are discretized near the boundary the influence of the divergence free condition for vorticity to the overall accuracy is studied results on the cubic driven cavity flow at reynolds number 500 and 3200 are shown and compared with that of the mac scheme

a review of the decomposition method and some recent results for nonlinear equations abstract the decomposition method can be an effective procedure for solution of nonlinear and or stochastic continous time dynamical systems without usual restrictive assumptions this paper is intended as a convenient tutorial review of the method 1

equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations abstract we consider the solution of partial differential equations for initial boundary conditions using the decomposition method the partial solutions obtained from the seperate equations for the highest ordered linear operator terms are shown to be identical when the boundary conditions are general and asymptotically equal when the boundary conditions in one independent variable are independet of other variables

solution of the navier stokes equation i abstract the decomposition method is applied to the navier stokes equation to provide a solution without linearization or discretization or restrictions on randomness

local distortion of m conformal mappings a conformal mapping in a plane domain locally maps circles to circles more generally quasiconformal mappings locally map circles to ellipses of bounded distortion in this work we study the corresponding situation for solutions to stein weiss systems in the n 1 d euclidean space this class of solutions is a transformation of a subset of monogenic locally quasiconformal mappings with nonvanishing jacobian in the theoretical part of this work we prove that an m conformal mapping locally maps the unit hypersphere onto explicitly characterized hyperellipsoids and vice versa then we discuss quasiconformal radial mappings and their relations with the cauchy kernel and p monogenic mappings this is followed by the consideration of quadratic m conformal mappings in the applications part of this work we provide the reader with some plot examples that demonstrate the effectiveness of our approach

application of the adomian decomposition method for two dimensional parabolic equation subject to nonstandard boundary specifications in this work we are concerned with the solution of a nonlocal boundary value problem an approach is presented for solving the two dimensional parabolic partial differential equation subject to integral boundary specifications the main objective is to propose an alternative method of solution one not based on finite difference methods or finite element schemes or spectral techniques the aim of the present paper is to investigate the application of the adomian decomposition method for solving the two dimensional linear parabolic partial differential equation with nonlocal boundary specifications replacing the classical boundary conditions the adomian decomposition method is used by many researchers to investigate several scientific applications and requires less work if compare with the traditional techniques the introduction of this idea as will be discussed not only provides the solution in a series form but it also guarantees considerable saving of the calculations volume the solutions will be handle more easily quickly and elegantly without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions in this approach the solution is found in the form of a convergent power series with easily computed components the adomian decomposition scheme is easy to program in applied problems and provides immediate and convergent solutions without any need for linearization or discretization to give a clear overview of the methodology we have selected illustrative example

a two step adomian decomposition method the adomian decomposition method adm is a nonnumerical method for solving a wide variety of functional equations and usually gets the solution in a series form in this paper we propose an efficient modification of adm namely two step adomian decomposition method tsadm that will facilitate the calculations we then conduct a comparative study between the tsadm and previous methods with the help of several illustrative examples the results obtained indicate that the tsadm is effective and promising

an explicit divergence free dg method for incompressible flow abstract we present an explicit divergence free dg method for incompressible flow based on velocity formulation only an h div conforming and globally divergence free finite element space is used for the velocity field and the pressure field is eliminated from the equations by design the resulting ode system can be discretized using any explicit time stepping methods we use the third order strong stability preserving runge kutta method in our numerical experiments our spatial discretization produces the identical velocity field as the divergence conforming dg method of cockburn et al 2007 based on a velocity pressure formulation when the same dg operators are used for the convective and viscous parts due to the global nature of the divergence free constraint and its interplay with the boundary conditions it is very hard to construct local bases for our finite element space here we present a key result on the efficient implementation of the scheme by identifying the equivalence of the mass matrix inversion of the globally divergence free finite element space to a standard hybrid mixed poisson solver hence in each time step a hybrid mixed poisson solver is used which reflects the global nature of the incompressibility condition in the actual implementation of this fully discrete scheme the pressure field is also computed via the hybrid mixed poisson solver hence the scheme can be interpreted as a velocity pressure formulation that treats the incompressibility constraint and pressure forces implicitly but the viscous and convective part explicitly since we treat viscosity explicitly for the navier stokes equation our method shall be best suited for unsteady high reynolds number flows so that the cfl constraint is not too restrictive

an application of the adm to seven order sawada kotara equations we implemented the adomian decomposition method for short adm for approximating the solution of the seventh order sawada kotera for short ssk and a lax s seventh order kdv for short lskdv equations by using this scheme explicit exact solution is calculated we obtain the exact solitary wave solutions and numerical solutions of the lskdv and ssk equations for the initial conditions the numerical solutions are compared with the known analytical solutions their remarkable accuracy are finally demonstrated for the both seven order equations

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105