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The purpose of this paper is to find a semi‐analytic solution to the fractional oscillator equations. In this paper, the authors apply the modified differential transform method to find approximate analytical solutions to fractional oscillators.

The modified differential transform method is used to obtain the solutions of the systems. This approach rests on the recently developed modification of the differential transform method. Some examples are given to illustrate the ability and reliability of the modified differential transform method for solving fractional oscillators.

The main conclusion is that the proposed method is a good way for solving such problems. The results are compared with those obtained by the fourth‐order Runge‐Kutta method. It is shown that the results reveal that the modified differential transform method in many instances gives better results.

The paper demostrates that a hybrid method of differential transform method, Laplace transform and Padé approximations provides approximate solutions of the oscillatory systems.

The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain.

The proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations.

The fractional derivative of Lagrange polynomial is a big hurdle in classical DQM. To overcome this problem, the authors represent the Lagrange polynomial in terms of shifted Legendre polynomial. They construct a transformation matrix which transforms the Lagrange polynomial into shifted Legendre polynomial of arbitrary order. Then, they obtain the new weighting coefficients matrices for space fractional derivatives by shifted Legendre polynomials and use these in conversion of a non-linear fractional partial differential equation into a system of fractional ordinary differential equations. Convergence analysis for the proposed method is also discussed.

Many engineers can use the presented method for solving their time and space fractional non-linear partial differential equation models. To the best of the authors’ knowledge, the differential quadrature method has never been extended or implemented for non-linear time and space fractional partial differential equations.

Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an efficient and convenient numerical algorithm for space-time fractional differential equations of the Fokker–Planck type.

The main idea of the presented algorithm is to combine polynomials function approximation and fractional differential operator matrices to reduce the studied complex equations to easily solved algebraic equations.

Based on Taylor basis, simple and useful fractional differential operator matrices of alternative Legendre polynomials can be quickly obtained, by which the studied space-time fractional partial differential equations can be transformed into easily solved algebraic equations. Numerical examples and error date are presented to illustrate the accuracy and efficiency of this technique.

Various numerical methods are proposed in complex steps and are computationally expensive. However, the advantage of this paper is its convenient technique, i.e. using the simple fractional differential operator matrices of polynomials, numerical solutions can be quickly obtained in high precision. Presented numerical examples can also indicate that the technique is feasible for this kind of fractional partial differential equations.

The purpose of this paper is to propose an approximate method for solving a fractional population growth model in a closed system. The fractional derivatives are described in the Caputo sense.

The approach is based on the differential transform method. The solutions of a fractional model equation are calculated in the form of convergent series with easily computable components.

The diagonal Padé approximants are effectively used in the analysis to capture the essential behavior of the solution.

Illustrative examples are included to demonstrate the validity and applicability of the technique.

The purpose of this paper is to introduce stability analysis for the initial value problem for the fractional Schrödinger differential equation: Equation 1 in a Hilbert space H with a self‐adjoint positive definite operator A under the condition |α(s)|<M₁/s^1/2 and the first order of accuracy difference scheme for the approximate solution of this initial value problem.

The paper considers the stability estimates for the solution of this problem and the stability estimate for the approximate solution of first order of accuracy difference scheme of this problem.

The paper finds the stability for the fractional Schrödinger differential equation and the first order of accuracy difference scheme of that equation by applications to one‐dimensional fractional Schrödinger differential equation with nonlocal boundary conditions and multidimensional fractional Schrödinger differential equation with Dirichlet conditions.

The paper is a significant work on stability of fractional Schrödinger differential equation and its difference scheme presenting some numerical experiments which resulted from applying obtained theorems on several mixed fractional Schrödinger differential equations.

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.

This paper aims to present a numerical solution of non‐linear Burger's equation using differential quadrature method based on sinc functions.

Sinc Differential Quadrature Method is used for space discretization and four stage Runge‐Kutta algorithm is used for time discretization. A rate of convergency analysis is also performed for shock‐like solution. Numerical stability analysis is performed.

Sinc Differential Quadrature Method generates more accurate solutions of Burgers' equation when compared with the other methods.

This combination, Sinc Differential Quadrature and Runge‐Kutta of order four, has not been used to obtain numerical solutions of Burgers' equation.

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.

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 simulate numerical solutions of nonlinear Burgers' equation with two well‐known problems in order to verify the accuracy of the cubic B‐spline differential quadrature methods.

Cubic B‐spline differential quadrature methods have been used to discretize the Burgers' equation in space and the resultant ordinary equation system is integrated via Runge‐Kutta method of order four in time. Numerical results are compared with each other and some former results by calculating discrete root mean square and maximum error norms in each case. A matrix stability analysis is also performed by determining eigenvalues of the coefficient matrices numerically.

Numerical results show that differential quadrature methods based on cubic B‐splines generate acceptable solutions of nonlinear Burgers' equation. Constructing hybrid algorithms containing various basis to determine the weighting coefficients for higher order derivative approximations is also possible.

Nonlinear Burgers' equation is solved by cubic B‐spline differential quadrature methods.