On Exact Solutions of Second Order Nonlinear Ordinary Differential Equations

Zraiqat, Amjed; AL-Hwawcha, Laith K.

doi:10.4236/am.2015.66087

Cited by 8 publications

(3 citation statements)

References 4 publications

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“…The curves serve as visual representations of how the properties of GARCH reverberate through the bilinear models, providing valuable insights into the complex interactions within the data and informing our understanding of financial market behavior. See [15][16][17][18].…”

Section: Asymmetric and Symmetric Garch Modelsmentioning

confidence: 99%

Comparative Analysis of Bilinear Time Series Models with Time-Varying and Symmetric GARCH Coefficients: Estimation and Simulation

Abu Hammad,

Alkhateeb,

Laiche

et al. 2024

Symmetry

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This paper makes a significant contribution by focusing on estimating the coefficients of a sample of non-linear time series, a subject well-established in the statistical literature, using bilinear time series. Specifically, this study delves into a subset of bilinear models where Generalized Autoregressive Conditional Heteroscedastic (GARCH) models serve as the white noise component. The methodology involves applying the Klimko–Nilsen theorem, which plays a crucial role in extracting the asymptotic behavior of the estimators. In this context, the Generalized Autoregressive Conditional Heteroscedastic model of order (1,1) noted that the GARCH (1,1) model is defined as the white noise for the coefficients of the example models. Notably, this GARCH model satisfies the condition of having time-varying coefficients. This study meticulously outlines the essential stationarity conditions required for these models. The estimation of coefficients is accomplished by applying the least squares method. One of the key contributions lies in utilizing the fundamental theorem of Klimko and Nilsen, to prove the asymptotic behavior of the estimators, particularly how they vary with changes in the sample size. This paper illuminates the impact of estimators and their approximations based on varying sample sizes. Extending our study to include the estimation of bilinear models alongside GARCH and GARCH symmetric coefficients adds depth to our analysis and provides valuable insights into modeling financial time series data. Furthermore, this study sheds light on the influence of the GARCH white noise trace on the estimation of model coefficients. The results establish a clear connection between the model characteristics and the nature of the white noise, contributing to a more profound understanding of the relationship between these elements.

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Section: Asymmetric and Symmetric Garch Modelsmentioning

confidence: 99%

Comparative Analysis of Bilinear Time Series Models with Time-Varying and Symmetric GARCH Coefficients: Estimation and Simulation

Abu Hammad,

Alkhateeb,

Laiche

et al. 2024

Symmetry

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“…General results of the solvability and uniqueness were inferred by different methods such as the energy method, upper lower method and the Fadeo Galerkin methods. The later one is regarded one of the most important methods that were mainly developed in the 1960s, but they are still powerful tools today to deal with nonlinear evolution equations, especially those who are modeled by non-classical boundary conditions that consist of integral conditions [10][11][12][13]. Non-local and integral partial differential equations are used to solve a vast range of current physics and technology challenges [14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Solvability of Nonlinear Wave Equation with Nonlinear Integral Neumann Conditions

Batiha¹,

Chebana²,

Oussaeif³

et al. 2023

Int. J. Anal. Appl.

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In this paper, we examine a nonlinear hyperbolic equation with a nonlinear integral condition. In particular, we prove the existence and the uniqueness of the linear problem by the Fadeo Galerkin method, and by applying an iterative process to some significant results obtained for the linear problem, the existence and the uniqueness of the weak solution for the nonlinear problem are additionally examined.

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“…During the last 30 years there have been done a lot of progress in finding solutions to nonlinear ODEs and PDEs. The progress is mostly made by using different methods like the Prelle-Singer method [4], Abel's equations [5] [6], the new Jacobi elliptic functions [7] [8], the old Jacobi elliptic functions [9] [10], a new method [11], revised methods [12], Jacobi elliptic function expansion method [13], expo-elliptic functions [14].…”

Section: Introductionmentioning

confidence: 99%

Four New Examples of the Non-Elementary Expo-Elliptic Functions That Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

Stensland¹

2022

JAMP

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In this paper, we define four new examples of the non-elementary expo-elliptic functions. This is an exponential function whose exponent is the product of a real number and the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel's methods, described by Armitage and Eberlein. We will study some of the second-order nonlinear ODEs, especially those that exhibit limit cycles, and systems of nonlinear ODEs that these functions are giving solutions to.

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On Exact Solutions of Second Order Nonlinear Ordinary Differential Equations

Abstract: In this paper, a new approach for solving the second order nonlinear ordinary differential equation y'' + p(x; y)y' = G(x; y) is considered. The results obtained by this approach are illustrated by examples and show that this method is powerful for this type of equations.

Cited by 8 publications

References 4 publications

Comparative Analysis of Bilinear Time Series Models with Time-Varying and Symmetric GARCH Coefficients: Estimation and Simulation

Comparative Analysis of Bilinear Time Series Models with Time-Varying and Symmetric GARCH Coefficients: Estimation and Simulation

Solvability of Nonlinear Wave Equation with Nonlinear Integral Neumann Conditions

Four New Examples of the Non-Elementary Expo-Elliptic Functions That Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

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