General viscosity implicit midpoint rule for nonexpansive mapping

In this work, we suggest a general viscosity implicit midpoint rule for nonexpansive mapping in the framework of Hilbert space. Further, under the certain conditions imposed on the sequence of parameters, strong convergence theorem is proved by the sequence generated by the proposed iterative scheme, which, in addition, is the unique solution of the variational inequality problem. Furthermore, we provide some applications to variational inequalities, Fredholm integral equations, and nonlinear evolution equations and give a numerical example to justify the main result. The results presented in this work may be treated as an improvement, extension and refinement of some corresponding ones in the literature.

Draining of Water Tank using Runge-Kutta Methods

The use of water tanks as a tool for storing water before being distributed for daily use has become a widely used system today. Among the attempts to develop a water distribution system is optimization in terms of system and operating costs. In this study, four methods of the Runge Kutta method are the Implicit such as Explicit Euler method, Implicit Euler method, Implicit Midpoint Rule, Runge Kutta Fourth-order method are used and compared with the exact solution method. The method will be compared in terms of accuracy and efficiency in solving differential equations based on set parameters for optimum design of water tank. The accuracy and efficiency of each method can be determined based on error graph. At the end of the study, numerical results obtained indicate that the Implicit Midpoint Rule provides greater stability and accuracy for the fixed stepsize given compared to other numerical methods.

The Generalized Viscosity Implicit Midpoint Rule for Nonexpansive Mappings in Banach Space

This paper constructs the generalized viscosity implicit midpoint rule for nonexpansive mappings in Banach space. It obtains strong convergence conclusions for the proposed algorithm and promotes the related results in this field. Moreover, this paper gives some applications. Finally, the paper gives six numerical examples to support the main results.

On New Picard-Mann Iterative Approximations with Mixed Errors for Implicit Midpoint Rule and Applications

In order to solve (partial) differential equations, implicit midpoint rules are often employed as a powerful numerical method. The purpose of this paper is to introduce and study a class of new Picard-Mann iteration processes with mixed errors for the implicit midpoint rules, which is different from existing methods in the literature, and to analyze the convergence and stability of the proposed method. Further, some numerical examples and applications to optimal control problems with elliptic boundary value constraints are considered via the new Picard-Mann iterative approximations, which shows that the new Picard-Mann iteration process with mixed errors for the implicit midpoint rule of nonexpansive mappings is brand new and more effective than other related iterative processes.