A new approach for demonstrating the global stability of ordinary differential equations is given. It is shown that if the curvature of solutions is bounded on some set, then any nonconstant orbits that remain in the set, must contain points that lie some minimum distance apart from each other. This is used to establish a negative-criterion for periodic orbits. This is extended to give a method of proving an equilibrium to be globally stable. The approach can also be used to rule out the sudden appearance of large-amplitude periodic orbits.
{"title":"On Using Curvature to Demonstrate Stability","authors":"C. McCluskey","doi":"10.1155/2008/745242","DOIUrl":"https://doi.org/10.1155/2008/745242","url":null,"abstract":"A new approach for demonstrating the global stability of ordinary differential equations is given. It is shown that if the curvature of solutions is bounded on some set, then any nonconstant orbits that remain in the set, must contain points that lie some minimum distance apart from each other. This is used to establish a negative-criterion for periodic orbits. This is extended to give a method of proving an equilibrium to be globally stable. The approach can also be used to rule out the sudden appearance of large-amplitude periodic orbits.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2008 1","pages":"1-7"},"PeriodicalIF":0.0,"publicationDate":"2008-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2008/745242","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64178938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The homotopy analysis method (HAM) is applied to solve linear and nonlinear fractional partial differential equations (fPDEs). The fractional derivatives are described by Caputo's sense. Series solutions of the fPDEs are obtained. A convergence theorem for the series solution is also given. The test examples, which include a variable coefficient, inhomogeneous and hyperbolic-type equations, demonstrate the capability of HAM for nonlinear fPDEs.
{"title":"Series Solutions of Time-Fractional PDEs by Homotopy Analysis Method","authors":"O. Abdulaziz, I. Hashim, A. Saif","doi":"10.1155/2008/686512","DOIUrl":"https://doi.org/10.1155/2008/686512","url":null,"abstract":"The homotopy analysis method (HAM) is applied to solve linear and nonlinear fractional partial differential equations (fPDEs). The fractional derivatives are described by Caputo's sense. Series solutions of the fPDEs are obtained. A convergence theorem for the series solution is also given. The test examples, which include a variable coefficient, inhomogeneous and hyperbolic-type equations, demonstrate the capability of HAM for nonlinear fPDEs.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2008 1","pages":"686512"},"PeriodicalIF":0.0,"publicationDate":"2008-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2008/686512","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64177460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present work, we consider a class of nonlinear optimal control problems, which can be called “optimal control problems in mechanics.” We deal with control systems whose dynamics can be described by a system of Euler-Lagrange or Hamilton equations. Using the variational structure of the solution of the corresponding boundary-value problems, we reduce the initial optimal control problem to an auxiliary problem of multiobjective programming. This technique makes it possible to apply some consistent numerical approximations of a multiobjective optimization problem to the initial optimal control problem. For solving the auxiliary problem, we propose an implementable numerical algorithm.
{"title":"Optimal Control of Mechanical Systems","authors":"V. Azhmyakov","doi":"10.1155/2007/18735","DOIUrl":"https://doi.org/10.1155/2007/18735","url":null,"abstract":"In the present work, we consider a class of nonlinear optimal control problems, which can be called “optimal control problems in mechanics.” We deal with control systems whose dynamics can be described by a system of Euler-Lagrange or Hamilton equations. Using the variational structure of the solution of the corresponding boundary-value problems, we reduce the initial optimal control problem to an auxiliary problem of multiobjective programming. This technique makes it possible to apply some consistent numerical approximations of a multiobjective optimization problem to the initial optimal control problem. For solving the auxiliary problem, we propose an implementable numerical algorithm.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"32 1","pages":"1-16"},"PeriodicalIF":0.0,"publicationDate":"2007-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2007/18735","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64140198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies the existence of global solutions to the initial-boundary value problem for some nonlinear degenerate wave equations by means of compactness method and the potential well idea. Meanwhile, we investigate the decay estimate of the energy of the global solutions to this problem by using a difference inequality.
{"title":"Global Existence and Asymptotic Behavior of Solutions for a Class of Nonlinear Degenerate Wave Equations","authors":"Y. Ye","doi":"10.1155/2007/19685","DOIUrl":"https://doi.org/10.1155/2007/19685","url":null,"abstract":"This paper studies the existence of global solutions to the initial-boundary value problem for some nonlinear degenerate wave equations by means of compactness method and the potential well idea. Meanwhile, we investigate the decay estimate of the energy of the global solutions to this problem by using a difference inequality.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"8 1","pages":"1-9"},"PeriodicalIF":0.0,"publicationDate":"2007-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2007/19685","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64140105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The stochastic finite element method (SFEM) is employed for solving stochastic one-dimension time-dependent differential equations with random coefficients. SFEM is used to have a fixed form of linear algebraic equations for polynomial chaos coefficients of the solution process. Four fixed forms are obtained in the cases of stochastic heat equation with stochastic heat capacity or heat conductivity coefficients and stochastic wave equation with stochastic mass density or elastic modulus coefficients. The relation between the exact deterministic solution and the mean of solution process is numerically studied.
{"title":"Stochastic Finite Element Technique for Stochastic One-Dimension Time-Dependent Differential Equations with Random Coefficients","authors":"M. Saleh, I. El-Kalla, M. Ehab","doi":"10.1155/2007/48527","DOIUrl":"https://doi.org/10.1155/2007/48527","url":null,"abstract":"The stochastic finite element method (SFEM) is employed for solving stochastic one-dimension time-dependent differential equations with random coefficients. SFEM is used to have a fixed form of linear algebraic equations for polynomial chaos coefficients of the solution process. Four fixed forms are obtained in the cases of stochastic heat equation with stochastic heat capacity or heat conductivity coefficients and stochastic wave equation with stochastic mass density or elastic modulus coefficients. The relation between the exact deterministic solution and the mean of solution process is numerically studied.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2007 1","pages":"1-16"},"PeriodicalIF":0.0,"publicationDate":"2007-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2007/48527","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64150222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper investigates the local existence of the nonnegative solution and the finite time blow-up of solutions and boundary layer profiles of diffusion equations with nonlocal reaction sources; we also study the global existence and that the rate of blow-up is uniform in all compact subsets of the domain, the blow-up rate of |u(t)|∞ is precisely determined.
{"title":"Uniform Blow-Up Rates and Asymptotic Estimates of Solutions for Diffusion Systems with Nonlocal Sources","authors":"Zhoujin Cui, Zuodong Yang","doi":"10.1155/2007/87696","DOIUrl":"https://doi.org/10.1155/2007/87696","url":null,"abstract":"This paper investigates the local existence of the nonnegative solution and the finite time blow-up of solutions and boundary layer profiles of diffusion equations with nonlocal reaction sources; we also study the global existence and that the rate of blow-up is uniform in all compact subsets of the domain, the blow-up rate of |u(t)|∞ is precisely determined.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2007 1","pages":"91-106"},"PeriodicalIF":0.0,"publicationDate":"2007-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2007/87696","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64158996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper presents models of car dynamics with varying complexity. Joint coordinates and homogenous transformations are used to model the motion of a car. Having formulated the models of the car, we discuss the influence of the complexity of the model on numerical efficiency of integrating the equations describing car dynamics. Methods with both constant and adaptive step size have been applied. The results of numerical calculations are presented and conclusions are formulated.
{"title":"Numerical Effectiveness of Models and Methods of Integration of the Equations of Motion of a Car","authors":"M. Szczotka, S. Tengler, S. Wojciech","doi":"10.1155/2007/49157","DOIUrl":"https://doi.org/10.1155/2007/49157","url":null,"abstract":"The paper presents models of car dynamics with varying complexity. \u0000Joint coordinates and homogenous transformations are used to model the motion \u0000of a car. Having formulated the models of the car, we discuss the influence of the \u0000complexity of the model on numerical efficiency of integrating the equations \u0000describing car dynamics. Methods with both constant and adaptive step size have \u0000been applied. The results of numerical calculations are presented and conclusions \u0000are formulated.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2007 1","pages":"1-13"},"PeriodicalIF":0.0,"publicationDate":"2007-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2007/49157","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64150472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We implement a relatively new analytical technique, the variational iteration decomposition method (VIDM), for solving the eighth-order boundary value problems. The proposed method is an elegant combination of variational iteration method and decomposition method. The analytical results of the equations have been obtained in terms of convergent series with easily computable components. Numerical work is given to check the efficiency of the method. Comparisons are made to confirm the reliability and accuracy of the technique. The technique can be used as an alternative for solving nonlinear boundary value problems.
{"title":"Variational Iteration Decomposition Method for Solving Eighth-Order Boundary Value Problems","authors":"M. Noor, S. Mohyud-Din","doi":"10.1155/2007/19529","DOIUrl":"https://doi.org/10.1155/2007/19529","url":null,"abstract":"We implement a relatively new analytical technique, the variational iteration decomposition method (VIDM), for solving the eighth-order boundary value problems. The proposed method is an elegant combination of variational iteration method and decomposition method. The analytical results of the equations have been obtained in terms of convergent series with easily computable components. Numerical work is given to check the efficiency of the method. Comparisons are made to confirm the reliability and accuracy of the technique. The technique can be used as an alternative for solving nonlinear boundary value problems.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2007 1","pages":"1-16"},"PeriodicalIF":0.0,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2007/19529","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64140009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the first-order nonhomogenous wave equation. We extend the convolution theorem into a general case with a double convolution as the nonhomogenous term. The uniqueness and continuity of the solution are proved and we provide some examples in order to validate our results.
{"title":"A Note on Wave Equation and Convolutions","authors":"A. Kılıçman, H. Eltayeb","doi":"10.1155/2007/49251","DOIUrl":"https://doi.org/10.1155/2007/49251","url":null,"abstract":"We study the first-order nonhomogenous wave equation. We extend the convolution theorem into a general case with a double convolution as the nonhomogenous term. The uniqueness and continuity of the solution are proved and we provide some examples in order to validate our results.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2007 1","pages":"1-13"},"PeriodicalIF":0.0,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2007/49251","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64150530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A perturbation-theoretic analysis of rectifier circuits is presented. The governing differential equation of the half-wave rectifier with capacitor filter is analyzed by expanding the output voltage as a Taylor series with respect to an artificially introduced parameter in the nonlinearity of the diode characteristic as is done in quantum theory. The perturbation parameter introduced in the analysis is independent of the circuit components as compared to the method presented by multiple scales. The various terms appearing in the perturbation series are then modeled in the form of an equivalent circuit. This model is subsequently used in the analysis of full-wave rectifier. Matlab simulation results are included which confirm the validity of the theoretical formulations. Perturbation analysis acts a helpful tool in analyzing time-varying systems and chaotic systems.
{"title":"A perturbation-based model for rectifier circuits","authors":"V. B. Vats, H. Parthasarathy","doi":"10.1155/DENM/2006/32675","DOIUrl":"https://doi.org/10.1155/DENM/2006/32675","url":null,"abstract":"A perturbation-theoretic analysis of rectifier circuits is presented. The governing differential equation of the half-wave rectifier with capacitor filter is analyzed by expanding the output voltage as a Taylor series with respect to an artificially introduced parameter in the nonlinearity of the diode characteristic as is done in quantum theory. The perturbation parameter introduced in the analysis is independent of the circuit components as compared to the method presented by multiple scales. The various terms appearing in the perturbation series are then modeled in the form of an equivalent circuit. This model is subsequently used in the analysis of full-wave rectifier. Matlab simulation results are included which confirm the validity of the theoretical formulations. Perturbation analysis acts a helpful tool in analyzing time-varying systems and chaotic systems.","PeriodicalId":30100,"journal":{"name":"Differential Equations and Nonlinear Mechanics","volume":"2006 1","pages":"1-13"},"PeriodicalIF":0.0,"publicationDate":"2006-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/DENM/2006/32675","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64875379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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