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To provide a new proof of convergence of the Adomian decomposition series for solving nonlinear ordinary and partial differential equations based upon a thorough examination of the historical milieu preceding the Adomian decomposition method.

Develops a theoretical background of the Adomian decomposition method under the auspices of the Cauchy‐Kovalevskaya theorem of existence and uniqueness for solution of differential equations. Beginning from the concepts of a parametrized Taylor expansion series as previously introduced in the Murray‐Miller theorem based on analytic parameters, and the Banach‐space analog of the Taylor expansion series about a function instead of a constant as briefly discussed by Cherruault et al., the Adomian decompositions series and the series of Adomian polynomials are found to be a uniformly convergent series of analytic functions for the solution u and the nonlinear composite function f(u). To derive the unifying formula for the family of classes of Adomian polynomials, the author develops the novel notion of a sequence of parametrized partial sums as defined by truncation operators, acting upon infinite series, which induce these parametrized sums for simple discard rules and appropriate decomposition parameters. Thus, the defining algorithm of the Adomian polynomials is the difference of these consecutive parametrized partial sums.

The four classes of Adomian polynomials are shown to belong to a common family of decomposition series, which admit solution by recursion, and are derived from one unifying formula. The series of Adomian polynomials and hence the solution as computed as an Adomian decomposition series are shown to be uniformly convergent. Furthermore, the limiting value of the mth Adomian polynomial approaches zero as the index m approaches infinity for the prerequisites of the Cauchy‐Kovalevskaya theorem. The novel truncation operators as governed by discard rules are analogous to an ideal low‐pass filter, where the decomposition parameters represent the cut‐off frequency for rearranging a uniformly convergent series so as to induce the parametrized partial sums.

This paper unifies the notion of the family of Adomian polynomials for solving nonlinear differential equations. Further it presents the new notion of parametrized partial sums as a tool for rearranging a uniformly convergent series. It offers a deeper understanding of the elegant and powerful Adomian decomposition method for solving nonlinear ordinary and partial differential equations, which are of paramount importance in modeling natural phenomena and man‐made device performance parameters.

The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more accurate results.

The authors proposed a method by using the Chebyshev wavelets in conjunction with differential quadrature technique. The operational matrices for the method are derived, constructed and used for the solution of nonlinear fractional partial differential equations.

The operational matrices contain many zero entries, which lead to the high efficiency of the method and reasonable accuracy is achieved even with less number of grid points. The results are in good agreement with exact solutions and more accurate as compared to Haar wavelet method.

Many engineers can use the presented method for solving their nonlinear fractional models.

The purpose of this study is to obtain the numerical scheme of finding the numerical solutions of arbitrary order partial differential equations subject to the initial and boundary conditions.

The authors present a novel Green-Haar approach for the family of fractional partial differential equations. The method comprises a combination of Haar wavelet method with the Green function. To handle the nonlinear fractional partial differential equations the authors use Picard technique along with Green-Haar method.

The results for some numerical examples are documented in tabular and graphical form to elaborate on the efficiency and precision of the suggested method. The obtained results by proposed method are compared with the Haar wavelet method. The method is better than the conventional Haar wavelet method, for the tested problems, in terms of accuracy. Moreover, for the convergence of the proposed technique, inequality is derived in the context of error analysis.

The authors present numerical solutions for nonlinear Burger’s partial differential equations and two-term partial differential equations.

Engineers and applied scientists may use the present method for solving fractional models appearing in applications.

The purpose of this paper is to use the polynomial differential quadrature method (PDQM) to find the numerical solutions of some Burgers'‐type nonlinear partial differential equations.

The PDQM changed the nonlinear partial differential equations into a system of nonlinear ordinary differential equations (ODEs). The obtained system of ODEs is solved by Runge‐Kutta fourth order method.

Numerical results for the nonlinear evolution equations such as 1D Burgers', coupled Burgers', 2D Burgers' and system of 2D Burgers' equations are obtained by applying PDQM. The numerical results are found to be in good agreement with the exact solutions.

A comparison is made with those which are already available in the literature and the present numerical schemes are found give better solutions. The strong point of these schemes is that they are easy to apply, even in two‐dimensional nonlinear problems.

Aims to analyse unique deformation properties of textile materials in terms of basic mechanical properties. Models fabric deformation as a nonlinear dynamical system so that a fabric can be completely specified in terms of its mechanical behaviour under general boundary conditions. Fabric deformation is dynamically analogous to waves travelling in a fluid. A localized two‐dimensional deformation evolves through the fabric to form a three‐dimensional drape or fold configuration. The nonlinear differential equations arising in the analysis of fabric deformation belong to the Klein‐Gordon family of equations which becomes the sine‐Gordon equation in three dimensions. The sine‐Gordon equation has its origins in the study of Bäcklund Transformations in differential geometry. Describes fabric deformation as a series of transformations of surfaces, defined in terms of curvature parameters using Gaussian representation of surfaces. By considering a deformed fabric as a two‐dimensional surface, algebraically constructs analytical solutions of fabric deformation by solving the sine‐Gordon Equation. The theory of Bäcklund Transformations is used to transform a trivial solution into a series of solitary wave solutions. These analytical expressions describing the curvature parameters of a surface represent actual solutions of fabric dynamical systems.

This paper seeks to develop, propose and validate, through a series of presentable examples, a comprehensive high‐precision and ultra‐fast computing concept for solving stiff ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN).

The core of the concept developed in this paper is a straight‐forward scheme that we call “nonlinear adaptive optimization (NAOP)”, which is used for a precise template calculation for solving any (stiff) nonlinear ODEs through CNN processors.

One of the key contributions of this work (this is a real breakthrough) is to demonstrate the possibility of mapping/transforming different types of nonlinearities displayed by various classical and well‐known oscillators (e.g. van der Pol‐, Rayleigh‐, Duffing‐, Rössler‐, Lorenz‐, and Jerk‐ oscillators, just to name a few) unto first‐order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN‐templates. Furthermore, in case of PDEs solving, the same concept also allows a mapping unto first‐order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDEs in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultra‐fast solver of stiff differential equations (both ODEs and PDEs). This clearly enables a CNN‐based, real‐time, ultra‐precise, and low‐cost Computational Engineering. As proof of concept a well‐known prototype of stiff equations (van der Pol) has been considered; the corresponding precise CNN‐templates are derived to obtain precise solutions of this equation.

This paper contributes to the enrichment of the literature as the relevant state‐of‐the‐art does not provide a systematic and robust method to solve nonlinear ODEs and/or nonlinear PDEs using the CNN‐paradigm. Further, the “NAOP” concept developed in this paper has been proven to perform accurate and robust calculations. This concept is not based on trial‐and‐error processes as it is the case for various classes of optimization methods/tools (e.g. genetic algorithm, particle swarm, neural networks, etc.). The “NAOP” concept developed in this frame does significantly contribute to the consolidation of CNN as a universal and ultra‐fast solver of nonlinear differential equations (both ODEs and PDEs). An implantation of the concept developed is possible even on embedded digital platforms (e.g. field‐programmable gate array (FPGA), digital signal processing (DSP), graphics processing unit (GPU), etc.); this opens a broad range of applications. On‐going works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting PDE models such as Navier Stokes, Schrödinger, Maxwell, etc.

The purpose of this paper is to use He's Exp‐function method (EFM) to construct solitary and soliton solutions of the nonlinear evolution equation.

This technique is straightforward and simple to use and is a powerful method to overcome some difficulties in the nonlinear problems.

This method is developed for searching exact traveling wave solutions of the nonlinear partial differential equations. The EFM presents a wider applicability for handling nonlinear wave equations.

The paper shows that EFM, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations. Application of EFM to Fitzhugh‐Nagumo equation illustrates its effectiveness.

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. This paper aims to propose a new Yang-Abdel-Aty-Cattani (YAC) fractional operator with a non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, reduced differential transform method (RDTM) and residual power series method (RPSM) taking the fractional derivative as YAC operator sense.

This study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense.

This study has expressed the solutions in terms of Mittag-Leffler functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this study, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

The population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit.

The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R_(y,s) (x, t) and r_(y,s) (x, t) reproducing kernel functions.

Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models.

Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers.

The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability.

Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest.

This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.