Efficient Decomposition Shooting Method for Solving Third-Order Boundary Value Problems

The present paper proposes a mathematical method to numerically treat a class of third-order linear Boundary Value Problems (BVPs). This method is based on the combination of the Adomian Decomposition Method (ADM) and, the modified shooting method. A complete derivation of the proposed method has been provided, in addition to its numerical implementation and, validation via the utilization of the Runge-Kutta method and, other existing methods. The method has been applied to diverse test problems and turned out to perform remarkably. Lastly, the simulated numerical results have been graphically illustrated and, also supported by some absolute error comparison tables.

Mode decomposition of nonlinear eigenvalue problems and application in flow stability

Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.

Analytic decomposition and numerical procedure for solving the singular boundary value problem arising in viscous flows

An efficient analytical decomposition technique was presented for solving the singular nonlinear boundary value problem arising in viscous flow when the Crocco variable was introduced. The approximate analytical solution may be represented in terms of a rapid convergent power series with elegantly computable terms. The reliability and efficiency of the approximate solutions were verified by numerical ones in the literature. The approximate analytical solutions can be successfully applied to give the values of skin friction coefficient.

ANALYSIS OF WAVEGUIDE PROBLEMS USING A RELAXED ITERATIVE DOMAIN DECOMPOSITION METHOD COMBINED WITH MULTIFRONTAL ALGORITHM

In this paper, an absorbing Fictitious Boundary Condition (FBC) is presented to generate an iterative Domain Decomposition Method (DDM) for analyzing waveguide problems.The relaxed algorithm is introduced to improve the iterative convergence. And the matrix equations are solved using the multifrontal algorithm. The resulting CPU time is greatly reduced.Finally, a number of numerical examples are given to illustrate its accuracy and efficiency.

Key Exchange Protocol Based on Tensor Decomposition Problem

The hardness of tensor decomposition problem has many achievements, but limited applications in cryptography, and the tensor decomposition problem has been considered to have the potential to resist quantum computing. In this paper, we firstly proposed a new variant of tensor decomposition problem, then two one-way functions are proposed based on the hard problem. Secondly we propose a key exchange protocol based on the one-way functions, then the security analysis, efficiency, recommended parameters and etc. are also given. The analyses show that our scheme has the following characteristics: easy to implement in software and hardware, security can be reduced to hard problems, and it has the potential to resist quantum computing.Besides the new key exchange can be as an alternative comparing with other classical key protocols.

On the Application of Adomian Decomposition Method to Special Equations in Physical Sciences

The current manuscript makes use of the prominent iterative procedure, called the Adomian Decomposition Method (ADM), to tackle some important special differential equations. The equations of curiosity in this study are the singular equations that arise in many physical science applications. Thus, through the application of the ADM, a generalized recursive scheme was successfully derived and further utilized to obtain closed-form solutions for the models under consideration. The method is, indeed, fascinating as respective exact analytical solutions are accurately acquired with only a small number of iterations.

THE CONVERGENCE OF A DOMAIN DECOMPOSITION ALGORITHM FOR PARABOLIC PROBLEMS

At recent, Hourgat et gave a domain decomposition algorithm for elliptic problems which can be implemented in parallel. Many numerical experiments have illustrated its efficiency. In the present paper, we apply this algorithm to solve the discrete parabolic problems, analyse its convergence and show that its convergence rale is about (1 - 2p + σp2 ) which is nearly optimal and independent of the parameter τ, where σ τ O((1 +H )(1 + ln(H / h))2 ). 0 【 p 【 1 / σ,τ,h,H are the time step size, finite element parameter and subdomain diameter, respectively.

A new high-order spectral problem of the mKdV and its associated integrable decomposition

A new approach to formulizing a new high-order matrix spectral problem from a normal 2 × 2 matrix modified Korteweg-de Vries （mKdV） spectral problem is presented. It is found that the isospectral evolution equation hierarchy of this new higher-order matrix spectral problem turns out to be the well-known mKdV equation hierarchy. By using the binary nonlinearization method, a new integrable decomposition of the mKdV equation is obtained in the sense of Liouville. The proof of the integrability shows that r-matrix structure is very interesting,

CHEBYSHEV PSEUDOSPECTRAL DOMAIN DECOMPOSITION METHOD OF ONE－DIMENSIONAL ELLIPTIC PROBLEMS

This paper is devoted to the Chebyshev pseudospectral domain decomposition method of one-dimensional elliptic problems,it is easily applied to complex geometry.The approximate accuracy can be increased by increasing the order of approximation in fixed number of subdomains,rather than by resorting to a further partitioning.The stability and the convergence of this method are proved.

Adomian Decomposition Method with Green’s Function for Solving Tenth-Order Boundary Value Problems

In this paper, the Adomian decomposition method with Green’s function (Standard Adomian and Modified Technique) is applied to solve linear and nonlinear tenth-order boundary value problems with boundary conditions defined at any order derivatives. The numerical results obtained with a small amount of computation are compared with the exact solutions to show the efficiency of the method. The results show that the decomposition method is of high accuracy, more convenient and efficient for solving high-order boundary value problems.

Numerical Treatment of Initial Value Problems of Nonlinear Ordinary Differential Equations by Duan-Rach-Wazwaz Modified Adomian Decomposition Method

We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.

Adomian Decomposition Method for Solving Goursat's Problems

In this paper, Goursat’s problems for: linear and nonlinear hyperbolic equations of second-order, systems of nonlinear hyperbolic equations and fourth-order linear hyperbolic equations in which the attached conditions are given on the characteristics curves are transformed in such a manner that the Adomian decomposition method (ADM) can be applied. Some examples with closed-form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient.

A New m × m Matrix Kaup-Newell Spectral Problem and Its Associated Integrable Decomposition

A new m × m matrix Kaup-Newell spectral problem is constructed from a normal 2 × 2 matrix Kaup-Newell spectral problem, a new integrable decomposition of the Kaup-Newell equation is presented. Through this process, we find the structure of the r-matrix is interesting.

Recent Modifications of Adomian Decomposition Method for Initial Value Problem in Ordinary Differential Equations

In this paper, some modifications of Adomian decomposition method are presented for solving initial value problems in ordinary differential equations. Also, the restarted and two-step methods are applied to the problem. The effectiveness of the each modified is verified by several examples.

Vertical Decomposition Approach to Solve Single Stage Capacitated Warehouse Location Problem (SSCWLP)

Single Stage Capacitated Warehouse Location Problem (SSCWLP) has been attempted by few researchers in the past. These are Geoffrion and Graves [1], Sharma [2], Sharma [3] and Sharma and Berry [4]. In this paper we give a “vertical decomposition” approach to solve SSCWLP that uses Lagrangian relaxation. This way SSCWLP is broken into two versions of capacitated plant location problem (the CPLP_L and CPLP_R) by relaxing the flow balance constraints. For CPLP_R, we use well known Lagrangian relaxations given in literature (Christofides and Beasley [5] and Nauss [6]);and adopt them suitably for solving CPLP_L. We show theoretically in this paper that SSCWLP can be more efficiently solved by techniques of vertical decomposition developed in this paper than the method available in literature (Sharma and Berry [4]). Encouraging computational study is reported in this paper.

An Efficient Decomposition Method for Solving Bratu’s Boundary Value Problem

The purpose of this paper is to employ the Adomian Decomposition Method (ADM) and Restarted Adomian Decomposition Method (RADM) with new useful techniques to resolve Bratu’s boundary value problem by using a new integral operator. The solutions obtained in this way require the use of the boundary conditions directly. The obtained results indicate that the new techniques give more suitable and accurate solutions for the Bratu-type problem, compared with those for the ADM and its modification.

Inverse spectral decomposition using an I_p-norm constraint for the detection of close geological anomalies

An important application of spectral decomposition(SD)is to identify subsurface geological anomalies such as channels and karst caves,which may be buried in full-band seismic data.However,the classical SD methods including the wavelet transform(WT)are often limited by relatively low time-frequency resolution,which is responsible for false high horizonassociated space resolution probably indicating more geological structures,especially when close geological anomalies exist.To address this issue,we impose a constraint of minimizing an lp(0<p<1)norm of time-frequency spectral coefficients on the misfit derived by using the inverse WT and apply the generalized iterated shrinkage algorithm to invert for the optimal coefficients.Compared with the WT and inverse SD(ISD)using a typical l1-norm constraint,the modified ISD(MISD)using an lp-norm constraint can yield a more compact spectrum contributing to detect the distributions of close geological features.We design a 3 D synthetic dataset involving frequency-close thin geological anomalies and the other3 D non-stationary dataset involving time-close anomalies to demonstrate the effectiveness of MISD.The application of 4 D spectrum on a 3 D real dataset with an area of approximately 230 km2 illustrates its potential for detecting deep channels and the karst slope fracture zone.

Signal prediction based on empirical mode decomposition and artificial neural networks

In view of the usefulness of Empirical Mode Decomposition （EMD）, Artificial Neural Networks （ ANN）, and Most Relevant Matching Extension （MRME） methods in dealing with nonlinear signals, we pro- pose a new way of combining these methods to deal with signal prediction. We found the results of combining EMD with either ANN or MRME to have higher prediction precision for a time series than the result of using EMD alone.

A Comparison between Modified Sumudu Decomposition Method and Homotopy Perturbation Method

In this paper, we present a comparative study between the modified Sumudu decomposition method (MSDM) and homotopy perturbation method (HPM). The study outlines the important features of the two methods. The analysis will be explained by discussing the nonhomogeneous Kortewege-de Vries (KdV) problems.

Mathematical modelling of a biofilm: The Adomian decomposition method

A mathematical modelling by a biofilm under steady state conditions is discussed. The nonlinear differential Equations in biofilm reaction is solved using the Adomian decomposition method. Approximate analytical expressions for substrate concentration have been derived for all values of parameters δ and SL. These analytical results are compared with the available numerical results and are found to be in good agreement.