Showing papers on "Integrating factor published in 2009"
TL;DR: In this article, a direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations, which provides a more systematical and convenient handling of the solution process of non-linear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, and the mapping method.
Abstract: A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F -expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3 + 1 dimensional Jimbo–Miwa equation is treated, together with a Backlund transformation.
510 citations
TL;DR: In this article, sufficient conditions for the existence of mild solutions for some densely defined semilinear functional differential equations and inclusions involving the Riemann-Liouville fractional derivative are established.
Abstract: We establish sufficient conditions for the existence of mild solutions for some densely defined semilinear functional differential equations and inclusions involving the Riemann-Liouville fractional derivative. Our approach is based on the -semigroups theory combined with some suitable fixed point theorems.
240 citations
TL;DR: For a given second-order ordinary differential equation (ODE), several relationships among first integrals, integrating factors and λ-symmetries are studied in this paper, and an algorithm to find two functionally independent first integral is provided.
Abstract: For a given second-order ordinary differential equation (ODE), several relationships among first integrals, integrating factors and λ-symmetries are studied The knowledge of a λ-symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of partial differential equations If two nonequivalent λ-symmetries of the equation are known, then an algorithm to find two functionally independent first integrals is provided These methods include and complete other methods to find integrating factors or first integrals that are based on variational derivatives or in the Prelle–Singer method These results are applied to several ODEs that appear in the study of relevant equations of mathematical physics
105 citations
TL;DR: A novel hybrid method for the solution of ordinary and partial differential equations is presented, which creates trial solutions in neural network form using a scheme based on grammatical evolution.
99 citations
TL;DR: An extension of the differential transformation method (DTM), which is an analytical-numerical method for solving the fuzzy differential equation (FDE), is given and the concept of generalised H-differentiability is used.
90 citations
TL;DR: This paper demonstrates that transitions to epileptic dynamics via changes in system parameters are qualitatively the same as in the original model with delay, as well as demonstrating that the onset of epilepsy may arise due to regions of bistability.
Abstract: In this paper we describe how an ordinary differential equation model of corticothalamic interactions may be obtained from a more general system of delay differential equations. We demonstrate that transitions to epileptic dynamics via changes in system parameters are qualitatively the same as in the original model with delay, as well as demonstrating that the onset of epileptic activity may arise due to regions of bistability. Hence, the model presents in one unique framework, two competing theories for the genesis of epileptiform activity. Similarities between model transitions and clinical data are presented and we argue that statistics obtained from, and a parameter estimation of this model may be a potential means of classifying and predicting the onset and offset of seizure activity.
86 citations
TL;DR: A generalized concept of higher-order differentiability for fuzzy functions is presented and new definitions of solution to fuzzy differential equations are introduced.
Abstract: We firstly present a generalized concept of higher-order differentiability for fuzzy functions. Then we interpret th-order fuzzy differential equations using this concept. We introduce new definitions of solution to fuzzy differential equations. Some examples are provided for which both the new solutions and the former ones to the fuzzy initial value problems are presented and compared. We present an example of a linear second-order fuzzy differential equation with initial conditions having four different solutions.
80 citations
TL;DR: In this article, the Jacobi Last Multiplier of a second-order ordinary differential equation and its Lagrangian is used to determine the Lagrangians of the Painleve equations.
65 citations
01 Jan 2009
TL;DR: In this paper, the authors give an introduction to the symmetry approach to the problems of integrability and discuss a variety of the problems associated with the Symmetry Approach and modern trends.
Abstract: The goal of our lectures is to give an introduction to the symmetry approach to the problems of integrability. The basic concepts are discussed in the first two chapters where we give definitions and formulate statements in a simple way with complete proofs. In the other chapters we attempt to make a brief account of the results obtained in more than 20 years of the development and give references to original articles as well as to comprehensive review papers. We illustrate the achievements in the description and classification of integrable equations and discuss a variety of the problems associated with the Symmetry Approach and modern trends. Many people have contributed to the development of the Symmetry Approach. The main credits here have to be given to Alexei Shabat whose pioneer works often determined principal directions of the research. The major results in the solution of specific classification problems had been obtained by Serguei Svinolupov with his remarkable abilities to exercise and structuralize very complex algebraic computations. We would also like to mention significant contributions of Ravil Yamilov and Anatoli Zhiber and others. We are very grateful to the above-mentioned colleagues for numerous discussions and mutual collaborations which casted our understanding and vision of the Symmetry Approach to the testing and classification of integrable equations. In our lecture course we do not include the recent works of V. Adler, V. Marikhin, A. Shabat and R. Yamilov related to integrable chains, Bäcklund transformations and Lagrangian aspects of the Symmetry Approach (see [1] and references), nor the works of I. Habibullin devoted to a symmetry approach to initial-boundary problems for integrable equations (see for instance [23]).
61 citations
TL;DR: In this paper, a class of boundary value problems for fractional differential equations involving nonlinear integral conditions involving non-compactness is investigated, and the main tool used in their considerations is the technique associated with measures of noncompactity.
Abstract: The aim of this paper is to investigate a class of boundary value problem for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of noncompactness.
TL;DR: By using DTM, the method is extended for delay differential equations and is a reliable method that needs less work and does not require strong assumptions and linearization.
Abstract: Differential transform method (DTM) is extended for delay differential equations. By using DTM, we manage to obtain the numerical, analytical, and exact solutions of both linear and nonlinear equations. In comparison with the existing techniques, the DTM is a reliable method that needs less work and does not require strong assumptions and linearization.
TL;DR: In this article, the authors compare the limit of solutions to stochastic differential equation obtained by adding a noise of intensity e to the generalized Filippov notion of solutions of the ODE.
Abstract: When the right-hand side of an ordinary differential equation (ODE in short) is not Lipschitz, neither existence nor uniqueness of solutions remain valid. Nevertheless, adding to the differential equation a noise with nondegenerate intensity, we obtain a stochastic differential equation which has pathwise existence and uniqueness property. The goal of this short paper is to compare the limit of solutions to stochastic differential equation obtained by adding a noise of intensity e to the generalized Filippov notion of solutions to the ODE. It is worth pointing out that our result does not depend on the dimension of the space while several related works in the literature are concerned with the one dimensional case.
TL;DR: In this article, a method for finding general solutions of third-order nonlinear differential equations by extending the modified Prelle-Singer method is introduced, where the integrals of motion associated with the given equation are deduced, so that the general solution follows straightforwardly from these integrals.
Abstract: We introduce a method for finding general solutions of third-order nonlinear differential equations by extending the modified Prelle–Singer method. We describe a procedure to deduce all the integrals of motion associated with the given equation, so that the general solution follows straightforwardly from these integrals. The method is illustrated with several examples. Further, we propose a powerful method of identifying linearizing transformations. The proposed method not only unifies all the known linearizing transformations systematically but also introduces a new and generalized linearizing transformation. In addition to the above, we provide an algorithm to invert the non-local linearizing transformation. Through this procedure the general solution for the original nonlinear equation can be obtained from the solution of the linear ordinary differential equation.
TL;DR: This chapter summarizes several recent examples of work that has been done in immune modeling and discusses two specific examples of models based on DDEs that can be used to understand the dynamics of T cell regulation.
Abstract: Numerous aspects of the immune system operate on the basis of complex regulatory networks that are amenable to mathematical and computational modeling. Several modeling frameworks have recently been applied to simulating the immune system, including systems of ordinary differential equations, delay differential equations, partial differential equations, agent-based models, and stochastic differential equations. In this chapter, we summarize several recent examples of work that has been done in immune modeling and discuss two specific examples of models based on DDEs that can be used to understand the dynamics of T cell regulation.
TL;DR: Singular boundary value problems for ordinary differential equations model many real world phenomena ranging from different physics equations to biological, physiological, and medical processes as discussed by the authors, and have been extensively studied in the literature.
Abstract: Singular boundary value problems for ordinary differential equations model many real world phenomena ranging from different physics equations to biological, physiological, and medical processes 1–3 . This special issue places its emphasis on the study, theory, and applications of boundary value problems involving singularities. It includes some review articles such as 4 , equations with discontinuous nonlinearities 5 , boundary value problems with uncertainty 6 , fractional differential equations 7 , periodic or antiperiodic solutions 8 , and biological 9 or medical applications 10 . Different methods and techniques are used ranging from variational methods 11 to bifurcation techniques 12 . The editors aimed at a volume that may serve as a reference in the topic of the special issue and collect twenty five original and cutting-edge research articles by some of the top researchers in boundary value problems for ordinary differential equations worldwide and from many different countries Algeria, Austria, Bulgaria, China, Czech Republic, Greece, Iran, Ireland, Italy, Japan, New Zealand, Pakistan, Saudi Arabia, South Korea, Spain, USA . We would like to thank the authors for their contributions, the Editor-in-Chief of the journal, Professor Ravi P. Agarwal, and the Editorial Office of the journal for their support.
TL;DR: In this paper, the retract principle was applied to nonlinear solutions of a nonlinear nonlinear equation and sufficient conditions for existence of a positive solution were derived for the same class of solutions.
Abstract: The investigation of asymptotic behaviour of solutions of ordinary differential equations is often based on the application of the retract principle. Initially developed for ordinary differential equations, this technique was extended to other classes of equations. Not answered remains a problem concerning the possibility of extending this principle to neutral differential equations. The goal of the present paper is to partially fill this gap and develop a corresponding technique for the application of this principle. The applicability of the main result is illustrated on a nonlinear equation and sufficient conditions for existence of a positive solution are derived.
TL;DR: In this article, the authors considered the case of scalar second-order ODEs and obtained an extension of the Lie table for secondorder equations with two symmetries by splitting each complex Lie symmetry of the given ODE.
Abstract: A scalar complex ordinary differential equation can be considered as two coupled real partial differential equations, along with the constraint of the Cauchy–Riemann equations, which constitute a system of four equations for two unknown real functions of two real variables. It is shown that the resulting system possesses those real Lie symmetries that are obtained by splitting each complex Lie symmetry of the given complex ordinary differential equation. Further, if we restrict the complex function to be of a single real variable, then the complex ordinary differential equation yields a coupled system of two ordinary differential equations and their invariance can be obtained in a non-trivial way from the invariance of the restricted complex differential equation. Also, the use of a complex Lie symmetry reduces the order of the complex ordinary differential equation (restricted complex ordinary differential equation) by one, which in turn yields a reduction in the order by one of the system of partial differential equations (system of ordinary differential equations). In this paper, for simplicity, we investigate the case of scalar second-order ordinary differential equations. As a consequence, we obtain an extension of the Lie table for second-order equations with two symmetries.
TL;DR: In this article, the authors present reduction and reconstruction procedures for symmetric stochastic differential equations, similar to those available for ordinary differential equations for the Hamiltonian case, which is studied with special care and illustrated with several examples.
Abstract: We present reduction and reconstruction procedures for the solutions of symmetric stochastic differential equations, similar to those available for ordinary differential equations. In addition, we use the local tangent-normal decomposition, available when the symmetry group is proper, to construct local skew-product splittings in a neighborhood of any point in the open and dense principal orbit type. The general methods introduced in the first part of the paper are then adapted to the Hamiltonian case, which is studied with special care and illustrated with several examples. The Hamiltonian category deserves a separate study since in that situation the presence of symmetries implies in most cases the existence of conservation laws, mathematically described via momentum maps, that should be taken into account in the analysis.
TL;DR: In this paper, a direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations, including the NLS equation, a new Hamiltonian amplitude equation, the coupled Schrodinger-KdV equations and the Hirota-Maccari equations.
Abstract: By using some exact solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations. The method is implemented for the NLS equation, a new Hamiltonian amplitude equation, the coupled Schrodinger–KdV equations and the Hirota–Maccari equations. New exact complex solutions are obtained.
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TL;DR: The notion of mixed leadership in nonzero-sum differential games, where there is no fixed hierarchy in decision making with respect to the players, is introduced, and a complete set of equations which yield the controls in the mixed-leadership Stackelberg solution are obtained.
Abstract: This paper introduces the notion of mixed leadership in non-zero-sum differential games, where there is no fixed hierarchy in decision making with respect to the players Whether a particular player is leader or follower depends on the instrument variable s/he is controlling, and it is possible for a player to be both leader and follower, depending on the control variable The paper studies two-player open-loop differential games in this framework, and obtains a complete set of equations (differential and algebraic) which yield the controls in the mixed-leadership Stackelberg solution The underlying differential equations are coupled and have mixed boundary conditions The paper also discusses the special case of linear-quadratic differential games, in which case solutions to the coupled differential equations can be expressed in terms of solutions to coupled Riccati differential equations which are independent of the state trajectoryDedicated by the authors to Professor George Leitmann on the occasion of his 85th birthyear
TL;DR: In this paper, a new version of the homotopy perturbation method (NHPM) is introduced, which efficiently solves linear and non-linear ordinary differential equations, including Euler-Lagrange, Bernoulli and Ricatti differential equations.
Abstract: In this paper, we introduce a new version of the homotopy perturbation method (NHPM) that efficiently solves linear and non-linear ordinary differential equations. Several examples, including Euler-Lagrange, Bernoulli and Ricatti differential equations, are given to demonstrate the efficiency of the new method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010
TL;DR: In this article, it was shown that there always exists a smooth inverse integrating factor in a neighborhood of a limit cycle, obtaining a necessary and sufficient condition for the existence of an analytic one.
Abstract: In this paper we address the problem of existence of inverse integrating factors for an analytic planar vector field in a neighborhood of its nonwandering sets. It is proved that there always exists a smooth inverse integrating factor in a neighborhood of a limit cycle, obtaining a necessary and sufficient condition for the existence of an analytic one. This condition is expressed in terms of the Ecalle–Voronin modulus of the associated Poincare map. The existence of inverse integrating factors in a neighborhood of an elementary singularity is also established, and we give the first known examples of analytic vector fields in R not admitting a C inverse integrating factor in any neighborhood of either a limit cycle or a weak focus. Moreover, it is shown that a C inverse integrating factor of a C planar vector field must vanish identically on the polycycles which are limit sets of its flow, thereby solving a problem posed by Garcia and Shafer (J. Differential Equations 217 (2005) 363–376).
TL;DR: In this article, a function transformation is presented to obtain the general solutions of these sub-equations, and then new exact travelling wave solutions of the CKdV-MKdV equation and the CK-dV equations as applications of this transformation are obtained.
Abstract: Sub-equation methods are used for constructing exact travelling wave solutions of nonlinear partial differential equations. The key idea of these methods is to take full advantage of all kinds of special solutions of sub-equation, which is usually a nonlinear ordinary differential equation. We present a function transformation which not only gives us a clear relation among these sub-equation methods, but also can be used to obtain the general solutions of these sub-equations. And then new exact travelling wave solutions of the CKdV-MKdV equation and the CKdV equations as applications of this transformation are obtained, and the approach presented in this paper can be also applied to other nonlinear partial differential equations.
TL;DR: In this paper, a viability result for multidimensional, time dependent, stochastic differential equations driven by fractional Brownian motion with Hurst parameter 1 2 H 1, using pathwise approach, was proved.
TL;DR: It is shown that the Exp-function method with the help of symbolic computation provides a straightforward and very effective mathematical tool for solving nonlinear evolution equations in mathematical physics.
Abstract: In this paper, the Exp-function method is used to obtain general solutions of a first-order nonlinear ordinary differential equation with a fourth-degree nonlinear term Based on the first-order nonlinear ordinary equation and its general solutions, new and more general exact solutions with free parameters and arbitrary functions of the (2+1)-dimensional dispersive long wave equations are obtained, from which some hyperbolic function solutions are also derived when setting the free parameters as special values It is shown that the Exp-function method with the help of symbolic computation provides a straightforward and very effective mathematical tool for solving nonlinear evolution equations in mathematical physics
TL;DR: In this paper, two improved direct algebraic methods for constructing exact complex travelling wave solutions of nonlinear partial differential equations are presented, which are applied to NLS equation, and then new types exact complex solutions are obtained.
TL;DR: In this article, all solutions of a fourth-order nonlinear delay differential equation are shown to converge to zero or to oscillate, and the Riccati type techniques involving third-order linear differential equations are employed.
Abstract: All solutions of a fourth-order nonlinear delay differential equation are shown to converge to zero or to oscillate. Novel Riccati type techniques involving third-order linear differential equations are employed. Implications in the deflection of elastic horizontal beams are also indicated.
TL;DR: In this paper, the authors provide linearizability criteria for a class of systems of two third-order ODEs that is cubically nonlinear in the first derivative, by differentiating a system of second-order quadratically nonlinear ordinary differential equations and using the original system to replace the second derivatives.
Abstract: We provide linearizability criteria for a class of systems of two third-order ordinary differential equations that is cubically nonlinear in the first derivative, by differentiating a system of second-order quadratically nonlinear ordinary differential equations and using the original system to replace the second derivatives. The procedure developed splits into two cases: those for which the coefficients are constant and those for which they are variables. Both cases are discussed and examples given.