نشریه علوم دانشگاه خوارزمی
سال نهم شماره 1 (بهار 1388)

Some boundary value problems for the Cauchy-Riemann equation with non-local boundary conditions in several regions of plane have been investigated and solved by authors. In this paper, by making use of fundamental solutions of Cauchy-Riemann equations and by presenting analytic solutions to the above-mentioned boundary value problems, we try to present an analytic expression for the solution of Cauchy-Riemann equation in the first semi-quarter.

A direct method to determine numerical solutions of linear Volterra integro-differential equations is presented in this paper.. This method is based on block-pulse functions and its operational matrix. By using this approach, the integro-differential equation reduces to a linear lower triangular system of algebraic equations which can be solved easily. Some numerical examples are provided to illustrate accuracy and computational efficiency of the method.

In this note we introduce the notion of weak McCoy rings as a generalization of McCoy rings, and investigate their properties. Also we show that, if is a semi-commutative ring, then is weak McCoy if and only if is weak McCoy.

An efficient one-pot synthesis of 1,8-dioxo-octahydroxanthene is achieved through a condensation of arylaldehydes and 5,5-dimethyl-1,3-cyclohexanedione in the presence of N-bromosuccinimid. This method enjoys several advantages such as low cost, simple work up procedure and safe reaction conditions. Graphical Abstract Table 1. Optimization of reaction condition. Entry Amount of NBS Condition Yield (%)a 1 0 mol % Reflux/ethanol/20 h 0 2 2 mol % 50°C/ethanol/12 h tr 3 2 mol % Reflux/ethanol/14 h 15 4 5 mol % Reflux/ethanol/14 h 30 5 10 mol % Reflux/ethanol/14 h 55 6 15 mol % Reflux/ethanol/14 h 80 7 20 mol % Reflux/ethanol/14 h 94 8 20 mol % MW/DMAC/5 min 90 a Isolated yield.

The modules (rings) satisfying acc on certain submodules investigated in [2] and various important properties of Noetherian modules and rings can be generalized to modules and rings of this class. The present author introduced and developed the concept of modules (rings) satisfying dcc on certain submodules in [5] and [6]. In this paper we present a new characterization of rings satisfying dcc on certain submodules.

This paper presents an appropriate numerical method to solve nonlinear Fredholm integro-differential equations with time delay. Its approach is based on the Taylor expansion. This method converts the integro-differential equation and the given conditions into the matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Taylor expansion coefficients, so that the solution of this system yields the Taylor expansion coefficients of the solution function. Then, the performance of the method is evaluated with some examples