Search

In this paper, we apply the Bernstein collocation method to construct the solution set of the Sylvester matrix differential equation (Sy-MDE) which involves the Lyapunov matrix differential equation. The method depends on the collocation method and Bernstein polynomials. The main advantage of the proposed method is that by using this method Sy-MDE reduces to a linear system of algebraic equations which can be solved by using an appropriate iterative method. We analyze the error and give a theorem that bounds the error. We also give the residual correction procedure to estimate the error. By using the procedure, we obtain a new approximate solution, namely a corrected Bernstein collocation solution. To illustrate how the proposed method is applied, several examples are given. Numerical experiments show the effectiveness and accuracy of the method for solving such types of Sy-MDE. [ABSTRACT FROM AUTHOR]

The object of this paper is to study the sequence spaces : [DELTA]W(A, M,q,p,u,s), [DELTA][W.sub.0](A, M, [bar.q], p, u, s) and [DELTA][W.sub.[infinity]](A, M, [bar.q], p, u, s) associated with strongly Cezaro summable sequences and discuss some topological properties of these spaces and other related results., 1. DEFINITIONS AND NOTATIONS Let w denote the set of all sequences x = ([x.sub.n]), real or complex, let p = ([p.sub.n]),q = (qn) and [bar.q] = ([[bar.q].sub.n]) denote the [...]

In the present paper, a new technique Laplace residual power series (L-RPS) method is used for presenting approximate solutions (ASs) of the nonlinear fractional Fisher partial differential equation (NLFF-PDE), and then, we make a comparison between our results and that obtained via residual power series (RPS) method in the literature [1]. Moreover, we show that the 1st and 2nd coefficients of RPS-ASs obtained in [1] are identical to Eq. (2.12) and (2.16), respectively. In addition, we have corrected and improved the following error results seen in [1]: Firstly, we find that the 3rd, 4th and 5th coefficients of RPS-ASs are not agreement with Eqs. (2.20)–(2.22) that obtained in [1, Page 85]. Secondly, the authors in [1] misused used Caputo fractional derivative (C-FD) D2αin Eq. (2.18) on page 84 which leads the terms in Eqs. (2.19) and f3(x),f4(x)and f5(x)in Eqs. (2.20)-(2.22) are incorrect. Finally, we present the numerical and graphical solutions of fractional Fisher logistic model (FFLM) based on our new approach. To check the robustness, accuracy and efficiency of our proposed technique, we compute the absolute error for the ASs of L-RPS, sinc-collocation-finite difference (SCFD) and RPS methods.

We define the sequence spaces [wθ,M,p,u,Δ]σ , [wθ,M,p,u,Δ]σ0 , and [wθ,M,p,u,Δ]σ∞ which are defined by combining the concepts of Orlicz functions, invariant means, and lacunary convergence. We also study some inclusion relations and linearity properties of the above-mentioned spaces. These are generalizations of those defined and studied by Savaş and Rhoades in 2002 and some others before.

We define the sequence spaces [wθ,M,p,u,Δ]σ, [wθ,M,p,u,Δ]σ0, and [wθ,M,p,u,Δ]σ∞ which are defined by combining the concepts of Orlicz functions, invariant means, and lacunary convergence. We also study some inclusion relations and linearity properties of the above-mentioned spaces. These are generalizations of those defined and studied by Savaş and Rhoades in 2002 and some others before.