Solving Different Types of Differential Equations Using Modified and New Modified Adomian Decomposition Methods

The Modified Adomian Decomposition Method (MADM) is presented. A number of problems are solved to show the efficiency of the method. Further, a new solution scheme for solving boundary value problems with Neumann conditions is proposed. The scheme is based on the modified Adomian decomposition method and the inverse linear operator theorem. Several differential equations with Neumann boundary conditions are solved to demonstrate the high accuracy and efficiency of the proposed scheme.

DOMAIN DECOMPOSITION METHODS FOR SOLVING PDE's ON MULTI-PROCESSORS

In this paper, we discuss the parallel domain decomposition method(DDM)for solving PDE's on parallel computers. Three types of DDM: DDM with overlapping, DDM without overlapping and DDM with fictitious component are discussed in a uniform framework. The eonvergence of the asynchronous parallel algorithms based on DDM are discussed.

Sparse Modal Decomposition Method Addressing Underdetermined Vortex-Induced Vibration Reconstruction Problem for Marine Risers

When investigating the vortex-induced vibration(VIV)of marine risers,extrapolating the dynamic response on the entire length based on limited sensor measurements is a crucial step in both laboratory experiments and fatigue monitoring of real risers.The problem is conventionally solved using the modal decomposition method,based on the principle that the response can be approximated by a weighted sum of limited vibration modes.However,the method is not valid when the problem is underdetermined,i.e.,the number of unknown mode weights is more than the number of known measurements.This study proposed a sparse modal decomposition method based on the compressed sensing theory and the Compressive Sampling Matching Pursuit(Co Sa MP)algorithm,exploiting the sparsity of VIV in the modal space.In the validation study based on high-order VIV experiment data,the proposed method successfully reconstructed the response using only seven acceleration measurements when the conventional methods failed.A primary advantage of the proposed method is that it offers a completely data-driven approach for the underdetermined VIV reconstruction problem,which is more favorable than existing model-dependent solutions for many practical applications such as riser structural health monitoring.

Iterative Pure Source Transfer Domain Decomposition Methods for Helmholtz Equations in Heterogeneous Media

We extend the pure source transfer domain decomposition method(PSTDDM)to solve the perfectly matched layer approximation of Helmholtz scattering problems in heterogeneous media.We first propose some new source transfer operators,and then introduce the layer-wise and block-wise PSTDDMs based on these operators.In particular,it is proved that the solution obtained by the layer-wise PSTDDM in R2 coincides with the exact solution to the heterogeneous Helmholtz problem in the computational domain.Second,we propose the iterative layer-wise and blockwise PSTDDMs,which are designed by simply iterating the PSTDDM alternatively over two staggered decompositions of the computational domain.Finally,extensive numerical tests in two and three dimensions show that,as the preconditioner for the GMRES method,the iterative PSTDDMs are more robust and efficient than PSTDDMs for solving heterogeneous Helmholtz problems.

Deep Domain Decomposition Methods:Helmholtz Equation

This paper proposes a deep-learning-based Robin-Robin domain decomposition method(DeepDDM)for Helmholtz equations.We first present the plane wave activation-based neural network(PWNN),which is more efficient for solving Helmholtz equations with constant coefficients and wavenumber k than finite difference methods(FDM).On this basis,we use PWNN to discretize the subproblems divided by domain decomposition methods(DDM),which is the main idea of DeepDDM.This paper will investigate the number of iterations of using DeepDDM for continuous and discontinuous Helmholtz equations.The results demonstrate that:DeepDDM exhibits behaviors consistent with conventional robust FDM-based domain decomposition method(FDM-DDM)under the same Robin parameters,i.e.,the number of iterations by DeepDDM is almost the same as that of FDM-DDM.By choosing suitable Robin parameters on different subdomains,the convergence rate is almost constant with the rise of wavenumber in both continuous and discontinuous cases.The performance of DeepDDM on Helmholtz equations may provide new insights for improving the PDE solver by deep learning.

How can technology and efficiency alleviate the dilemma of economic growth and carbon emissions in China's industrial economy? A metafrontier decoupling decomposition analysis

This paper attempts to explore the decoupling relationship and its drivers between industrial economic increase and energy-related CO_(2) emissions(ICE). Firstly, the decoupling relationship was evaluated by Tapio index. Then, based on the DEA meta-frontier theory framework which taking into account the regional and industrial heterogeneity and index decomposition method, the driving factors of decoupling process were explored mainly from the view of technology and efficiency. The results show that during2000-2019, weak decoupling was the primary state. Investment scale expansion was the largest reason hindering decoupling process of industrial increase from ICE. Both energy saving and production technology achieved significant progress, which facilitated the decoupling process. Simultaneously, the energy technology gap and production technology gap among regions have been narrowed, and played a role in promoting decoupling process. On the contrary, both scale economy efficiency and pure technical efficiency have inhibiting effects on decoupling process. The former indicates that the scale economy of China's industry was not conducive to improve energy efficiency and production efficiency, while the latter indicates that resource misallocation problem may exist in both energy market and product market.

Theory and Applications of Distinctive Conformable Triple Laplace and Sumudu Transforms Decomposition Methods

This article presents some important results of conformable fractional partial derivatives.The conformable triple Laplace and Sumudu transform are coupled with the Adomian decomposition method where a new method is proposed to solve nonlinear partial differential equations in 3-space.Moreover,mathematical experiments are provided to verify the performance of the proposed method.A fundamental question that is treated in this work:is whether using the Laplace and Sumudu transforms yield the same results?This question is amply answered in the realm of the proposed applications.

The Monotone Robin-Robin Domain Decomposition Methods for the Elliptic Problems with Stefan-Boltzmann Conditions

This paper is concerned with the elliptic problems with nonlinear StefanBoltzmann boundary condition.By combining with the monotone method,the RobinRobin domain decomposition methods are proposed to decouple the nonlinear interface and boundary condition.The monotone properties are verified for both the multiplicative and the additive domain decomposition methods.The numerical results confirm the theoretical analysis.

On Time Fractional Partial Differential Equations and Their Solution by Certain Formable Transform Decomposition Method

This paper aims to investigate a new efficient method for solving time fractional partial differential equations.In this orientation,a reliable formable transform decomposition method has been designed and developed,which is a novel combination of the formable integral transform and the decomposition method.Basically,certain accurate solutions for time-fractional partial differential equations have been presented.Themethod under concern demandsmore simple calculations and fewer efforts compared to the existingmethods.Besides,the posed formable transformdecompositionmethod has been utilized to yield a series solution for given fractional partial differential equations.Moreover,several interesting formulas relevant to the formable integral transform are applied to fractional operators which are performed as an excellent application to the existing theory.Furthermore,the formable transform decomposition method has been employed for finding a series solution to a time-fractional Klein-Gordon equation.Over and above,some numerical simulations are also provided to ensure reliability and accuracy of the new approach.

Adomian Decomposition Method for Solving Boussinesq Equations Using Maple

This paper uses the Adomian Decomposition Method (ADM) to solve Boussinesq equations using Maple. The Boussinesq approximation for water waves is a weakly nonlinear and long-wave approximation in fluid dynamics. The approximation is named after Joseph Boussinesq, who developed it in response to John Scott Russell’s observation of a wave of translation (also known as solitary wave or soliton). Bossinesq’s article from 1872 introduced the equations that are now known as the Boussinesq equations. Numerical methods are commonly utilized to solve nonlinear equation systems. In this paper, we investigate a nonlinear singly perturbed advection-diffusion problem. Using the usual Adomian Decomposition Method, we formulate an approximate linear advection-diffusion problem and investigate several practical numerical approaches for solving it (ADM). The Adomian Decomposition Method (ADM) is a powerful tool for numerical simulations and approximation analytic solutions. The Adomian Decomposition Method (ADM) is used to solve nonlinear advection differential equations using Maple by illustrating numerous examples. The findings are presented in the form of tables and graphs for several examples. For various examples, the findings are presented in the form of tables and graphs. The difference between the precise and numerical solutions indicates the Maple program solution’s efficacy, as well as the ease and speed with which it was acquired.

The Natural Transform Decomposition Method for Solving Fractional Klein-Gordon Equation

In this paper, a coupling of the natural transform method and the Admoian decomposition method called the natural transform decomposition method (NTDM), is utilized to solve the linear and nonlinear time-fractional Klein-Gordan equation. The (NTDM), is introduced to derive the approximate solutions in series form for this equation. Solutions have been drawn for several values of the time power. To identify the strength of the method, three examples are presented.

Adomian Decomposition Method for Solving Time Fractional Burgers Equation Using Maple

In this paper, the Adomian decomposition method was used to solve the Time Fractional Burger equation using Mabel program. This method was applied to a number of examples of the Time Fractional Burger Equation. The obtained numerical results were presented in the form of tables and graphics. The difference between the exact solutions and the numerical solutions shows us the effectiveness of the solution using the Mabel program and that this method gave accurate results and was close to the exact solution, in addition to its ability to obtain the numerical solution quickly and efficiently using the Mabel program.

Application of the Modified Adomian Decomposition Method on a Mathematical Model of COVID-19

In this study, we constructed and analysed a mathematical model of COVID-19 in order to comprehend the transmission dynamics of the disease. The reproduction number (RC) was calculated via the next generation matrix method. We also used the Lyaponuv method to show the global stability of both the disease free and endemic equilibrium points. The results showed that the disease-free equilibrium point is globally asymptotically stable if RC RC > 1. We further used the Adomian decomposition method and the modified Adomian decomposition method to obtain the solutions of the model. Numerical analysis of the model was done using Sagemath 9.0 software.

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.

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.

Optimal Boundary Control Method for Domain Decomposition Algorithm

To study the domain decomposition algorithms for the equations of elliptic type, the method of optimal boundary control was used to advance a new procedure for domain decomposition algorithms and regularization method to deal with the ill posedness of the control problem. The determination of the value of the solution of the partial differential equation on the interface——the key of the domain decomposition algorithms——was transformed into a boundary control problem and the ill posedness of the control problem was overcome by regularization. The convergence of the regularizing control solution was proven and the equations which characterize the optimal control were given therefore the value of the unknown solution on the interface of the domain would be obtained by solving a series of coupling equations. Using the boundary control method the domain decomposion algorithm can be carried out.

Accurate simulations of pure-viscoacoustic wave propagation in tilted transversely isotropic media

Forward modeling of seismic wave propagation is crucial for the realization of reverse time migration(RTM) and full waveform inversion(FWI) in attenuating transversely isotropic media. To describe the attenuation and anisotropy properties of subsurface media, the pure-viscoacoustic anisotropic wave equations are established for wavefield simulations, because they can provide clear and stable wavefields. However, due to the use of several approximations in deriving the wave equation and the introduction of a fractional Laplacian approximation in solving the derived equation, the wavefields simulated by the previous pure-viscoacoustic tilted transversely isotropic(TTI) wave equations has low accuracy. To accurately simulate wavefields in media with velocity anisotropy and attenuation anisotropy, we first derive a new pure-viscoacoustic TTI wave equation from the exact complex-valued dispersion formula in viscoelastic vertical transversely isotropic(VTI) media. Then, we present the hybrid finite-difference and low-rank decomposition(HFDLRD) method to accurately solve our proposed pure-viscoacoustic TTI wave equation. Theoretical analysis and numerical examples suggest that our pure-viscoacoustic TTI wave equation has higher accuracy than previous pure-viscoacoustic TTI wave equations in describing q P-wave kinematic and attenuation characteristics. Additionally, the numerical experiment in a simple two-layer model shows that the HFDLRD technique outperforms the hybrid finite-difference and pseudo-spectral(HFDPS) method in terms of accuracy of wavefield modeling.

Semi-analytical investigation of heat transfer in a porous convective radiative moving longitudinal fin exposed to magnetic field in the presence of a shape-dependent trihybrid nanofluid

The thermal examination of a non-integer-ordered mobile fin with a magnetism in the presence of a trihybrid nanofluid(Fe_3O_4-Au-Zn-blood) is carried out. Three types of nanoparticles, each having a different shape, are considered. These shapes include spherical(Fe_3O_4), cylindrical(Au), and platelet(Zn) configurations. The combination approach is utilized to evaluate the physical and thermal characteristics of the trihybrid and hybrid nanofluids, excluding the thermal conductivity and dynamic viscosity. These two properties are inferred by means of the interpolation method based on the volume fraction of nanoparticles. The governing equation is transformed into a dimensionless form, and the Adomian decomposition Sumudu transform method(ADSTM) is adopted to solve the conundrum of a moving fin immersed in a trihybrid nanofluid. The obtained results agree well with those numerical simulation results, indicating that this research is reliable. The influence of diverse factors on the thermal overview for varying noninteger values of γ is analyzed and presented in graphical representations. Furthermore, the fluctuations in the heat transfer concerning the pertinent parameters are studied. The results show that the heat flux in the presence of the combination of spherical, cylindrical, and platelet nanoparticles is higher than that in the presence of the combination of only spherical and cylindrical nanoparticles. The temperature at the fin tip increases by 0.705 759% when the value of the Peclet number increases by 400%, while decreases by 11.825 13% when the value of the Hartman number increases by 400%.

Multi-Parameter Design and Optimization for the Decomposition Projective Method and Its Applications

In order to solve the electromagnetic problems on the large multi branch domains, the decomposition projective method(DPM) is generalized for multi subspaces in this paper. Furthermore multi parameters are designed for DPM, which is called the fast DPM(FDPM), and the convergence ratio of the above algorithm is greatly increased. The examples show that the iterative number of the FDPM with optimal parameters decreases much more, which is less than one third of the DPM iteration number. After studying the ...

DIRECTED HYPERGRAPH THEORY AND DECOMPOSITION CONTRACTION METHOD

A new branch of hypergraph theory-directed hyperaph theory and a kind of new methods-dicomposition contraction(DCP, PDCP and GDC) methods are presented for solving hypernetwork problems.lts computing time is lower than that of ECP method in several order of magnitude.