Muhammed I. Syam

AN EFFICIENT METHOD FOR SOLVING BRATU EQUATIONS

In this paper, we present a numerical technique for solving Bratu equation. It is based on the Laplace Adomain decomposition method which produces an implicit equation in two variables. We used the predictor corrector technique to trace the solution curve generated from this equation. Numerical results and conclusions will be presented.

Neural network assessment of herbal protection against chemotherapeutic-induced reproductive toxicity

The aim of this study is to assess the protective effects of Ginkgo bilobas (GB) extract against chemotherapeutic-induced reproductive toxicity using a data mining tool, namely Neural Network Clustering (NNC) on two types of data: biochemical & fertility indicators and Texture Analysis (TA) parameters. GB extract (1 g/kg/day) was given orally to male albino rats for 26 days. This period began 21 days before a single cisplatin (CIS) intraperitoneal injection (10 mg/kg body weight). GB given orally significantly restored reproductive function. Tested extract also notably reduced the CIS-induced reproductive toxicity, as evidenced by restoring normal morphology of testes. In GB, the attenuation of CIS-induced damage was associated with less apoptotic cell death both in the testicular tissue and in the sperms. CIS-induced alterations of testicular lipid peroxidation were markedly improved by the examined plant extract. NNC has been used for classifying animal groups based on the quantified biochemical & fertility indicators and microscopic image texture parameters extracted by TA. NNC showed the separation of two clusters and the distribution of groups among them in a way that signifies the dose-dependent protective effect of GB. The present study introduces the neural network as a powerful tool to assess both biochemical and histopathological data. We also show here that herbal protection against CIS-induced reproductive toxicity utilizing classic methodologies is validated using neural network analysis.

Design and application of nature inspired computing approach for nonlinear stiff oscillatory problems

Abstract In this paper, meta-heuristic intelligent approaches are developed for handling nonlinear oscillatory problems with stiff and non-stiff conditions. The mathematical modeling of these oscillators is accomplished using feed-forward artificial neural networks (ANNs) in the form of an unsupervised manner. The accuracy as well as efficiency of the model is subject to the tuning of adaptive parameters for ANNs that are highly stochastic in nature. These optimal weights are carried out with swarm intelligence and pattern search methods hybridized with an efficient local search technique based on constraints minimization known as active set algorithm. The proposed schemes are validated on various stiff and non-stiff variants of the oscillator. The significance, applicability and reliability of the proposed scheme are well established based on comparison made with the results of standard numerical solver.

Adomian decomposition method for approximating the solution of the Korteweg-deVries equation

In this paper, the Adomian decomposition method for solving the nonlinear Korteweg-deVries equation is implemented with appropriate initial conditions. We discuss the case when the problem has either one or more solitons. Some numerical examples are presented.

Numerical differentiation of implicitly defined curves

Abstract In this paper we develop a finite differencing device to calculate approximations of derivatives c′(0), c″(0),… of regular solution curves c : R ∋s→c(s)∈ R n of nonlinear systems of equations f(x)=0, f∈C k ( R n+1 , R n ) without having to compute points on the solution curve c ( s ). The derivative vectors c′(0), c″(0),… can be used in the numerical approximation of the solution set f −1 (0) to construct higher-order predictors to be used in the predictor–corrector continuation method.

An efficient implicit Runge–Kutta method for second order systems

Abstract We will consider the efficient implementation of a fourth order two stage implicit Runge–Kutta method to solve periodic second order initial value problems. To solve the resulting systems, we will use the factorization of the discretized operator. Such proposed factorization involves both complex and real arithmetic. The latter case is considered here. The resulting system will be efficient and small in size. It is one fourth the size of systems using normal implicit Runge–Kutta method. Numerical details and examples will also be presented to demonstrate the efficiency of the method.

Neuro-heuristic computational intelligence for solving nonlinear pantograph systems

We present a neuro-heuristic computing platform for finding the solution for initial value problems (IVPs) of nonlinear pantograph systems based on functional differential equations (P-FDEs) of different orders. In this scheme, the strengths of feed-forward artificial neural networks (ANNs), the evolutionary computing technique mainly based on genetic algorithms (GAs), and the interior-point technique (IPT) are exploited. Two types of mathematical models of the systems are constructed with the help of ANNs by defining an unsupervised error with and without exactly satisfying the initial conditions. The design parameters of ANN models are optimized with a hybrid approach GA–IPT, where GA is used as a tool for effective global search, and IPT is incorporated for rapid local convergence. The proposed scheme is tested on three different types of IVPs of P-FDE with orders 1–3. The correctness of the scheme is established by comparison with the existing exact solutions. The accuracy and convergence of the proposed scheme are further validated through a large number of numerical experiments by taking different numbers of neurons in ANN models.

Tau-Path Following Method for Solving the Riccati Equation with Fractional Order

A formulation for the fractional Legendre functions is constructed to find the solution of the fractional Riccati equation. The fractional derivative is described in the Caputo sense. The method is based on the Tau Legendre and path following methods. Theoretical and numerical results are presented. Analysis for the presented method is given.

Fractional-order Legendre operational matrix of fractional integration for solving the Riccati equation with fractional order

This paper is devoted to both theoretical and numerical study of Riccati equation with fractional order. A formulation to the fractional-order Legendre operational matrix of fractional integration is constructed. Existence and uniqueness results for the considered problem are provided and proved. The fractional derivative is described in the Caputo sense. Some numerical examples are discussed to demonstrate the efficiency and the accuracy of the proposed algorithm.

Evolutionary computational intelligence in solving a class of nonlinear Volterra-Fredholm integro-differential equations

In this paper, a stochastic computational intelligence technique for solving a class of nonlinear Volterra-Fredholm integro-differential equations with mixed conditions is presented. The strength of feed forward artificial neural networks is used to accurately model the integro-equation. Comparisons with the exact solution and other numerical techniques are presented to show the efficiency of the proposed method. Theoretical and numerical results are presented. Analysis for the presented method is given.