Narrow Results

In this paper, the closed form of solutions given for the following difference equation

In this paper, we give a numerical approximation to the Caputo–Fabrizio time fractional diffusion equation. The implicit finite differences method is applied to solve a time-fractional diffusion equation with this new fractional derivative. We present the stability and convergence analysis of the proposed numerical scheme. Some numerical problems will be presented to show the accuracy and effectiveness of the method.

In this paper, we obtain strong convergence results for asymptotically demicontractive and asymptotically hemicontractive mappings in hyperbolic spaces. We present our results in hyperbolic spaces. This class of spaces contains both linear and nonlinear spaces like CAT(0) spaces, $ℝ$-trees, Banach spaces and Hilbert spaces. Thus our results are not only novel but also much more general.

An Interesting result on a curious convergent series is proved by A. J. Kempner. In this short note we study and prove a result on another modification of Harmonic series.

Finding the solution of the absolute value equations (AVEs) has attracted much attention in recent years. In this paper, we propose and analyze two generalized accelerated overrelaxation (AOR) methods for solving AVEs $Ax-\left|x\right|=b$, where $A\in R^{n\times n}$ is an $M$-matrix. Furthermore, we discuss the convergence of the methods under some suitable assumptions. Numerical results are given to verify the effectiveness of our methods.

The harmonic series diverges. But, Kempner [A curious convergent series, Amer. Math Monthly 21 (1914) 48–50] proved that by removing those terms from the harmonic series whose denominators contain a digit $9$ anywhere, the resulting sub-series converges. Thus, Kempner allowed no 9’s at all, but in this short note, we allow 9’s with some restrictions and without tampering the convergence of the resultant series. Some results of our note generalize a recent note of Mukherjee and Sarkar [A short note on a curious convergent series, Asian-Eur. J. Math. 14(09) (2021) 2150158, doi:10.1142/S1793557121501588].

In this paper, we introduce a new iterative process for approximation fixed points. We show the stability of our proposed iteration process. We prove some weak and strong convergence theorems for generalized $\alpha$-nonexpansive mappings in the framework of uniformly convex Banach spaces. We also provide an example of a generalized $\alpha$-nonexpansive mapping to show numerically that our iteration is faster than various prominent iteration processes.

This paper describes two new iterative algorithms for determining absolute value equations. The algorithms are based on a splitting of the coefficient matrix. Moreover, we analyze the convergence effects of the presented algorithms via some theorems. Eventually, numerical tests are provided to confirm the credibility of our procedures.

The main purpose of this work is to connect the recently introduced iterative approximation scheme known as $KF$-iterative scheme with the class of generalized $(\alpha ,\beta )$-nonexpansive mappings. We consider the setting of a Banach space to prove our weak and strong convergence results. Once again, we prove that the class of $(\alpha ,\beta )$-nonexpansive mapping includes properly the classes of Suzuki nonexpansive, generalized $\alpha$-nonexpansive and as well Reich–Suzuki type nonexpansive mappings by providing a numerical example. Eventually, we prove that $KF$-iterative scheme of this example is essentially more efficient than the other well-known iterative scheme of the literature. The presented work is new and extends some other well-known results in the iterations theory.

In this paper, we use a smoothing-type algorithm in this paper to solve the AVE, which stands for the absolute value equation $Ax-\left|x\right|=b$, where $A$ is an arbitrary $n\times n$ real matrix and $b\in {ℝ}^n$. We reformulate AVE as a system of smooth equations and propose two new smoothing functions. We prove that the algorithm is well-defined when the singular value of $A$ exceeds one, and under the same assumption, the algorithm is convergent. We show the algorithm’s effectiveness with these two functions and compare it with some previously known functions.