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Series Solutions of Differential Equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
The Adomian decomposition method (ADM for short) was developed by an American mathematician and aerospace engineer of Armenian descent George Adomian (1922–1996). It suggests to seek a solution in the form y=∑n⩾0un,
Approximate Analytical Methods
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
The Adomian Decomposition Method (ADM) assumes that an equation's nonlinearity, and its solution, can be represented by convergent series. A formal solution representation is created and the terms sequentially determined. Typically, each term depends on all previously computed terms.
Analytical Solution for the Steady-State Heat Transfer Analysis of Porous Nonhomogeneous Material Structures
Published in Mohamed Thariq Hameed Sultan, Vishesh Ranjan Kar, Subrata Kumar Panda, Kandaswamy Jayakrishna, Advanced Composite Materials and Structures, 2023
Samarjeet Kumar, Vishesh Ranjan Kar, Bakhtiyar Alimovich Khudayarov
The heat transfer phenomena in nature can be categorized as linear (with some assumptions) and nonlinear. These phenomena can be modeled or expressed in the form of partial differential equations. Numerous methods exist for analyzing and expressing these problems. These methods are categorized as either analytical or numerical types. Some of the classical analytical methods used for solving these partial differential equations are the variable separation and homogenization method, integrating factor method, homotopy analysis method (HAM), and adomian decomposition method (ADM), among others [1–5]. The numerical methods include finite element method (FEM), finite difference method (FDM), boundary element method (BEM), meshless method, Heun’s method, Euler method, and Runge–Kutta method, among others [6–10]. The main problems associated with the analytical methods are that they can be solved for simple boundary and geometrically conditions. The introduction of complexity in the terms of material, geometry, and nonlinearity makes partial differential equations hard to solve. The solving process contains certain differential terms or integrations factor or some transform function that are not available or known. These terms or their values are fairly approximated to proceed further in numerical methods [11]. Sometimes solutions are referred to as semi-analytical, which means certain steps of partial differential equations are solved via the analytical or algebraic procedure and after that, numerical methods are used to approximate. The differential transform method (DTM) can be introduced as an analytical as well as a numerical method for solving integral equations and ordinary/partial differential equations. This method provides a good sense of continuation and gaining momentum among researchers due to its simplicity and pedagogical benefits [12]. This method provides analytical solutions in the form of a polynomial. It is different from the traditional high-order Taylor series, which is based on the symbolic computation of derivatives of functions. It can be regarded as an iterative procedure for obtaining the series solution of partial differential equations [3.13]. The obtained solution in the form of convergent series has easily computable components. The advantage associated with this method is less computational work along with a high convergence rate providing the accurate series solution.
Another way of solving a free boundary problem related to DCIS model
Published in Applicable Analysis, 2021
Jianrong Zhou, Yongzhi Xu, Heng Li
In this paper, we use different approach to solve the above problem [19]. Existence and uniqueness theorem of our problem (1)–(8) is obtained via Banach fixed point theorem in a constructed metric space. The proof of uniqueness is inspired by paper [20]. For numerical methods, although many numerical schemes such as finite element method and finite difference method seem to be used to obtain an approximate solution for such problem (1)–(8), there are several difficulties such as a mesh refinement, a stability condition, etc. To avoid these difficulties and find approximate solution, we introduce the Adomian decomposition method (ADM). It was developed by Adomian [21,22] and has been successfully applied to solve a wide class of linear and nonlinear problems [23–36].
Heat transfer and thermal analysis in a semi-spherical fin with temperature-variant thermal properties: an application of Probabilists’ Hermite collocation method
Published in Waves in Random and Complex Media, 2023
R. S. Varun Kumar, K. C. Jagadeesha, B. C. Prasannakumara
The above information exemplifies that thermal conductivity is temperature reliant in several engineering application domains. Accordingly, it is important to take into account the effects of temperature-dependent thermal properties when examining the fin in these kinds of situations. For such as well as in other engineering disciplines, thermal conductivity can be characterized by means of a power law and a linear temperature dependence. The dependence of thermal conductivity and heat transfer coefficient on temperature makes the problem extremely nonlinear and problematic to solve utilizing exact methods. In this context, several investigators utilized semi-analytical methods such as the differential transform method (DTM), perturbation method (PM), variational iteration method (VIM), homotopy analysis method (HAM), Adomian Decomposition Method (ADM), and homotopy perturbation method (HPM), to find the solutions for these types of equations. On the other hand, boundary value problems (BVPs) are crucial modeling tools for various physical phenomena including electromagnetism, boundary layer theory, diffusion process, heat transfer, and astronomy. For the solution of BVPs, a variety of numerical techniques have been proposed, and the collocation method is one of these that have been used extensively in recent years [32–34]. When using a suitable set of functions, also known as trial or basis functions, a collocation method is used to determine an approximate solution to an equation. At specific points in the range of interest known as collocation points, the approximate solution must fulfill the governing equation and its auxiliary conditions. In the case of linear and nonlinear differential equations, researchers have investigated various collocation methods including spectral, spline, and Chebyshev. However, several researchers used the collocation method with Hermite polynomials to solve various types of equations. Li and Zhang [35] implemented the effective probabilistic collocation method for inspecting the flow in permeable media. Luga et al. [36] used probabilists’ Hermite collocation method (PHCM) for solving the second-order nonlinear boundary value problems.