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The Painlev'e equations are second order ordinary differential equations which can be grouped into six families, namely Painlev'e equation I, II,?, VI. This paper presents the series solution of second Painlevé equation via optimal homotopy asymptotic method (OHAM). This approach is highly efficient and it controls the convergence of the approximate solution. Comparison of the obtained solution via OHAM is provided with those obtained by Homotopy Perturbation Method (HPM), Adomian Decomposition Method (ADM), Sinc-collocation and Runge-Kutta 4 methods. It is revealed that there is an excellent agreement between OHAM and other published data which confirm the effectiveness of the OHAM. [ABSTRACT FROM AUTHOR]

This paper presents an analytical solution of nonlinear differential equation of micro-structures subjected to electrostatic fields. The constitutive equation of such a model is a second order differential equation (ODE). The problem is solved when the assumption of linear deflection is considered. However, deflection of micro cantilevers in practical applications is nonlinear. Moreover, the constitutive ODE is stiff and various numerical algorithms used to solve it yield non-consistent numerical solutions. A deduction order method - Lie group symmetry is employed to reduce the order of the ODE. Although the resulting first order ODE has no symmetry that would guarantee an explicit close form solution, it enables an analytical formulation for the no-damping assumption only. The restoring force term in the first order ODE reveals the pull-in voltage as expressed in classical MEMS textbooks. It is shown that the numerical solution for the second order ODE and the reduced first order ODE are same. Finding any symmetry other than translation, scaling or rotation will enable the reduction of the first order ODE and thus, the formulation of an analytical solution to this highly non-linear problem. [ABSTRACT FROM AUTHOR]

The Painlev'e equations are second order ordinary differential equations which can be grouped into six families, namely Painlev'e equation I, II,…, VI. This paper presents the series solution of second Painlevé equation via optimal homotopy asymptotic method (OHAM). This approach is highly efficient and it controls the convergence of the approximate solution. Comparison of the obtained solution via OHAM is provided with those obtained by Homotopy Perturbation Method (HPM), Adomian Decomposition Method (ADM), Sinc-collocation and Runge-Kutta 4 methods. It is revealed that there is an excellent agreement between OHAM and other published data which confirm the effectiveness of the OHAM. doi:10.14456/WJST.2015.43

We consider a system of nonlinear Maxwell-type equations in the plane, which arises in the modeling of mesoscopic scale electromagnetic phenomena. After reduction to a quasi-linear equation and symmetrization, we obtain an explicit local description of the solutions in the form of power series. Subsequently, we apply general methods to show global existence of the solutions and to analyze their oscillations and asymptotic behavior. [ABSTRACT FROM AUTHOR]

Volterra's model for population growth in a closed system includes an integral term to indicate accumulated toxicity in addition to the usual terms of the logistic equation. In this research, a new numerical algorithm is introduced for solving this model. The proposed numerical approach is based on the modified Bessel function of the first kind and the collocation method. In this method, we aim to solve the problems on the semi-infinite domain without any domain truncation, variable transformation in basis functions and shifting the problem to a finite domain. Accordingly, we employ two different collocation approaches, one by computing through Volterra's population model in the integro-differential form and the other by computing by converting this model to an ordinary differential form. These methods reduce the solution of a problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of these methods, we compare the numerical results of the present methods with some well-known results in other to show that the new methods are efficient and applicable. [ABSTRACT FROM AUTHOR]

RS feedback models have been successful in explaining the observed phenomenon of clustering in autonomous oscillation in yeast, but current models do not include the biological reality of dynamical delay and do not have the related property of quorum sensing. Here an RS type ODE model for cell cycle feedback, including an explicit term modeling a chemical feedback mediating agent, is analyzed. New dynamics include population dependent effects: subcritical pitchfork bifurcations, and quorum sensing occur. The model suggests new experimental directions in autonomous oscillation in yeast. [ABSTRACT FROM AUTHOR]

In this paper we propose a method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. The proposed approach is based on an Unsupervised Combined Artificial Neural Networks (UCANN) method. Firstly, The trial solutions of the differential equations are written in the form of feed-forward neural networks containing adjustable parameters (the weights and biases); results are then optimized with the combined neural network. The proposed method is tested on series of Lane-Emden differential equations and the results are reported. Afterward, these results are compared with the solution of other methods demonstrating the efficiency and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Tau method for solving the singular Lane-Emden equation-a nonlinear ordinary differential equation on a semi-infinite interval. We applied collocation, Galerkin, and Tau methods for solving this problem, and according to the results, the solution of Tau method is the most accurate. The operational derivative and product matrices of the modified generalized Laguerre functions are presented. These matrices, in conjunction with the Tau method, are then utilized to reduce the solution of the Lane-Emden equation to that of a system of algebraic equations. We also present a comparison of this work with some well-known results and show that the present solution is highly accurate. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents a mathematical model for photo-excited carrier decay in a semiconductor. Due to the carrier trapping states and recombination centers in the bandgap, the carrier decay process is defined by the system of nonlinear differential equations. The system of nonlinear ordinary differential equations is approximated by linearized backward Euler scheme. Some a priori estimates of the discrete solution are obtained and the convergence of the linearized backward Euler method is proved. The identifiability analysis of the parameters of deep centers is performed and the fitting of the model to experimental data is done by using the genetic optimization algorithm. Results of numerical experiments are presented. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a collocation method using a new weighted orthogonal system on the half-line, namely the rational Gegenbauer functions, is introduced to solve numerically the third-order nonlinear differential equation, , where a is a constant parameter. This method solves the problems on semi-infinite domain without truncating it to a finite domain and transforming the domain of the problems to a finite domain. For , the equation is the well-known Blasius equation, which is a laminar viscous flow over a semi-infinite flat plate. We solve this equation by considering and compare the new results with the established results to show the efficiency and accuracy of the new method. [Copyright &y& Elsevier]

Abstract: A numerical technique for solving nonlinear ordinary differential equations on a semi-infinite interval is presented. We solve the Thomas–Fermi equation by the Sinc-Collocation method that converges to the solution at an exponential rate. This method is utilized to reduce the nonlinear ordinary differential equation to some algebraic equations. This method is easy to implement and yields very accurate results. [Copyright &y& Elsevier]

Abstract: This paper deals with the mathematical modeling of the mammary carcinoma–immune system competition elicited by an external stimulus represented by three different protocols of the triplex vaccine [C. De Giovanni, et al., Immunoprevention of HER-2/neu transgenic mammary carcinoma through an interleukin 12-engineered allogeneic cell vaccine, Cancer Research 64 (2004) 4001–4009]. The presented model is composed of nonlinear ordinary differential equations based on parameters and cell populations. A qualitative analysis of the asymptotic behavior of the model and numerical simulations are able to depict preclinical experiments on transgenic mice in tune with the SimTriplex model [F. Pappalardo, F. Castiglione, P.L. Lollini, S. Motta, Modelling and simulation of cancer immunoprevention vaccine, Bioinformatics 21 (2005) 2891–2897]. The results are of great interest both in the applied and theoretical sciences. [Copyright &y& Elsevier]

In this paper, we provide the collocation method for natural convection heat transfer equations embedded in porous medium which are of great importance in the design of canisters for nuclear waste disposal. This problem is a non-linear, three-point boundary value problem on semi-infinite interval. We use two orthogonal functions namely rational Gegenbauer and modified generalized Laguerre functions which are defined as basis functions in this approach and compare them together. We also present the comparison of these works with Runge-Kutta solution, moreover, in the graph of the ||Res||², we show that the present solutions are accurate and applicable. [ABSTRACT FROM AUTHOR]

Abstract: We solve a particular system of nonlinear ODEs defined on the two different components of the real line connected by the nonlinear contact condition We show that, for a prescribed power-law nonlinearity and using the solution , a self-similar solution to the porous medium equation in the two-component domain can be constructed. We provide the rigorous proof of a theorem in which the formula for the free boundary already presented in Filo and Pluschke (2009) is given. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of homoclinic solutions for the following class of second order ordinary differential equations (ODEs) where and are nonnegative continuous functions. Using Galerkin’s Method and assuming additional hypotheses on and , we show the existence of a homoclinic solution to problem . [Copyright &y& Elsevier]

Abstract: In this paper, a family of fourth-order Steffensen-type two-step methods is constructed to make progress in including Ren–Wu–Bi’s methods [H. Ren, Q. Wu, W. Bi, A class of two-step Steffensen type methods with fourth-order convergence, Appl. Math. Comput. 209 (2009) 206–210] and Liu–Zheng–Zhao’s method [Z. Liu, Q. Zheng, P. Zhao, A variant of Steffensens method of fourth-order convergence and its applications, Appl. Math. Comput. 216 (2010) 1978–1983] as its special cases. Its error equation and asymptotic convergence constant are deduced. The family provides the opportunity to obtain derivative-free iterative methods varying in different rates and ranges of convergence. In the numerical examples, the family is not only compared with the related methods for solving nonlinear equations, but also applied in the solution of BVPs of nonlinear ODEs by the finite difference method and the multiple shooting method. [Copyright &y& Elsevier]

Abstract: In this study, we investigate the magnetohydrodynamic (MHD) viscous flow due to a shrinking sheet by employing the homotopy perturbation method (HPM) and Padé approximation. The series solution of the governing nonlinear problem is developed. Generally, the truncated series solution is adequate only in a small region when the exact solution is not reached. We overcame this limitation by using the Padé techniques, which have the advantage in turning the polynomial approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. Comparison of the present solutions is made with the results obtained by other applied methods and excellent agreement is noted. [Copyright &y& Elsevier]

This paper describes a nonlinear oscillator governed by Duffing equation. An application for such a system is an oscillator with a nonlinear spring, for which the restoring spring force is given by --y+y³. In vibration theory such a spring is called hard spring, because the spring force per unit extension, -1+y², increases as the displacement increases. He's Homotopy Perturbation Method (HPM), which does not need small parameter in the equation is implemented for solving the Duffing equation. In continue results are compared with results extracted from an exact solution. The results reveal that the HPM is effective, convenient and quite accurate to systems of ordinary differential equations especially in the case of Duffing equation. It is predicted that the HPM can be found widely applicable in engineering. [ABSTRACT FROM AUTHOR]

Abstract: We consider the blow-up ODE problem , and a finite difference analogue, whose solution also blows up in finite time. In this paper, not only the convergence of the numerical blow-up time is proved but also the error estimate for the convergence is derived. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we propose a Lagrangian method for solving Lane–Emden equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on a Modified generalized Laguerre functions Lagrangian method. The method reduces the solution of this problem to the solution of a system of algebraic equations. We also present the comparison of this work with some well-known results and show that the present solution is acceptable. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we propose Sinc-Collocation method for solving Lane–Emden equation which is a nonlinear ordinary differential equation on a semi-infinite interval. It is found that Sinc procedure converges with the solution at an exponential rate. This method is utilized to reduce the computation of this problem to some algebraic equations. We also compare this solution with some well-known results and show that it is accurate. [Copyright &y& Elsevier]

Abstract: In this paper, the Cole–Hopf transform is used to construct exact solutions to a generalization of both the seventh-order Lax KdV equation (Lax KdV7) and the seventh-order Sawada–Kotera–Ito KdV equation (Sawada–Kotera–Ito KdV7 ) with forcing term. [Copyright &y& Elsevier]

Abstract: In this paper we propose, a collocation method for solving the Blasius equation. The Blasius equation is a third-order nonlinear ordinary differential equation. This approach is based on a rational scaled generalized Laguerre function collocation method. We also present the comparison of this work with some well-known results and show that the present solution is accurate. [Copyright &y& Elsevier]

Abstract: Lane–Emden equation is a nonlinear singular equation in the astrophysics that corresponds to the polytropic models. In this paper, a pseudospectral technique is proposed to solve the Lane–Emden type equations on a semi-infinite domain. The method is based on rational Legendre functions and Gauss–Radau integration. The method reduces solving the nonlinear ordinary differential equation to solve a system of nonlinear algebraic equations. The comparison of the results with the other numerical methods shows the efficiency and accuracy of this method. [Copyright &y& Elsevier]

Abstract: In this Letter we propose a pseudospectral method for solving Thomas–Fermi equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on rational Chebyshev pseudospectral method. This method reduces the solution of this problem to the solution of a system of algebraic equations. Comparison with some numerical solutions shows that the present solution is highly accurate. [Copyright &y& Elsevier]

Abstract: In this paper, the extended tanh method is used to construct generalized soliton solutions, periodic solutions and rational solutions for the Sawada–Kotera and Lax equations with forcing term. [Copyright &y& Elsevier]