Showing papers on "Exact differential equation published in 2012"
TL;DR: In this article, a method for finding exact solutions of nonlinear differential equations is considered and modifications of the method are discussed, showing that the method is one of the most effective approaches for solving nonlinear problems.
569 citations
TL;DR: In this paper, the (G'/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative.
Abstract: In this paper, the (G'/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann—Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota—Satsuma coupled KdV equations and the time-fractional fifth-order Sawada—Kotera equation. As a result, some new exact solutions for them are successfully established.
237 citations
01 Jan 2012
TL;DR: This chapter shows how to trigger the various methods using a variety of applications pointing, where necessary, to problems that may arise, and gives examples of how to implement a nonsmooth forcing term, switching behavior, and problems that include sudden jumps in the dependent variables.
Abstract: Both Runge-Kutta and linear multistep methods are available to solve initial value problems for ordinary differential equations in the R packages deSolve and deTestSet. Nearly all of these solvers use adaptive step size control, some also control the order of the formula adaptively, or switch between different types of methods, depending on the local properties of the equations to be solved. We show how to trigger the various methods using a variety of applications pointing, where necessary, to problems that may arise. For instance, many practical applications involve discontinuities. As the integration routines assume that a solution is sufficiently differentiable over a time step, handing such discontinuities requires special consideration. We give examples of how we can implement a nonsmooth forcing term, switching behavior, and problems that include sudden jumps in the dependent variables. Since much computational efficiency can be gained by using the correct method for a particular problem, we end this chapter by providing a few guidelines as to how the most efficient solution method for a particular problem can be found.
167 citations
TL;DR: The resulting nonlinear equation for the evolution of weakly nonlinear hydrodynamic disturbances on a static cosmological background with self-focusing in a two-dimensional nonlinear Schrodinger (NLS) equation was transformed to an ordinary differential equation, which depended only on one function ξ and can be solved.
138 citations
TL;DR: In this paper, the modified simple equation method is employed to construct the exact solutions involving parameters of nonlinear evolution equations via the (1+1)-dimensional modified KdV equation, and the ( 1+ 1)-dimensional reaction-diffusion equation.
Abstract: The modified simple equation method is employed to construct the exact solutions involving parameters of nonlinear evolution equations via the (1+1)-dimensional modified KdV equation, and the (1+1)-dimensional reaction-diffusion equation. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.
126 citations
TL;DR: In this paper, the authors used the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations, and used the improved (G ′ =G)-expansion function method to calculate the exact solutions to the time and space-fractional derivative foam drainage equation and the time-and space-fragments derivative nonlinear KdV equation.
Abstract: In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved (G ′ =G)-expansion function method to calculate the exact solutions to the time- and space-fractional derivative foam drainage equation and the time- and space-fractional derivative nonlinear KdV equation. This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.
117 citations
TL;DR: An exact upper bound is presented through the error analysis to solve the numerical solution of fractional differential equation with variable coefficient by using Haar wavelets and it is concluded that the method is convergent.
83 citations
TL;DR: In this article, the oscillation of the fractional differential equation was studied and the authors were concerned with the oscillations of the FDE with respect to the number of variables.
Abstract: In this article, we are concerned with the oscillation of the fractional differential equation
66 citations
TL;DR: In this article, the classification of exact solutions, including soliton, rational, and elliptic solutions, to the one-dimensional general improved Camassa Holm KP equation and KdV equation by the complete discrimination system for polynomial method was obtained.
Abstract: We obtain the classification of exact solutions, including soliton, rational, and elliptic solutions, to the one-dimensional general improved Camassa Holm KP equation and KdV equation by the complete discrimination system for polynomial method. In discussion, we propose a more general trial equation method for nonlinear partial differential equations with generalized evolution.
64 citations
TL;DR: In this paper, the Painleve analysis of (2+1)-variable coefficients Broer-Kaup (VCBK) equation is performed by the Weiss-Kruskal approach to check the painleve property.
TL;DR: In this article, a time-consistent optimal control problem is formulated and studied for a controlled linear ordinary differential equation with quadratic cost functional, and a notion of equilibrium control is introduced, which can be regarded as a timeconsistent solution to the original time-inconsistent problem.
Abstract: A time-inconsistent optimal control problem is formulated and studied for a controlled linear ordinary differential equation with quadratic cost functional. A notion of equilibrium control is introduced, which can be regarded as a time-consistent solution to the original time-inconsistent problem. Under certain conditions, we constructively prove the existence of such an equilibrium control which is represented via a forward ordinary differential equation coupled with a backward Riccati--Volterra integral equation. Our constructive approach is based on the introduction of a family of $N$-person non-cooperative differential games.
26 Sep 2012
TL;DR: In this article, the modified simple equation method was applied to find the exact solutions of nonlinear evolution equations via the (2+1)-dimensional Zakharov-Kuznetsev-Modified Equal-Width (ZK-MEW) equation and the (3+ 1)-dimensional Potential Yu-Toda-Sasa-Fukuyama (YTSF) equation.
Abstract: In this article, we apply the modified simple equation method for finding the exact solutions of nonlinear evolution equations via the (2+1)-dimensional Zakharov-Kuznetsev-Modified Equal-Width (ZK-MEW) equation and the (3+1)-dimensional Potential Yu-Toda-Sasa-Fukuyama (YTSF) equation. The applicability of this method for constructing these exact solutions is demonstrated.
TL;DR: In this paper, some basic results concerning the strict and nonstrict differential inequalities and existence of the maximal and minimal solutions are proved for a hybrid differential equation with linear perturbations of second type.
Abstract: In this paper, some basic results concerning the strict and nonstrict differential inequalities and existence of the maximal and minimal solutions are proved for a hybrid differential equation with linear perturbations of second type.
TL;DR: In this article, the modified simple equation method is applied to construct exact solutions of the modified equal width (MEW) equation and the Fisher equation, and the nonlinear Telegraph equation and Cahn-Allen equation.
TL;DR: In this article, the authors apply the Exp-function method to find exact solutions for two nonlinear partial differential equations (NPDE) and a nonlinear ordinary differential equation (NODE), namely, Cahn-Hilliard equation, Allen-Cahn equation and Steady-State equation, respectively.
TL;DR: Free and self-excited vibrations of conservative oscillators with polynomial nonlinearity are considered and the asymptotic averaging procedure for solving the perturbed strong non-linear differential equation is developed.
Abstract: Free and self-excited vibrations of conservative oscillators
with polynomial nonlinearity are considered. Mathematical model of the
system is a second-order differential equation with a nonlinearity of polynomial type, whose terms are of integer and/or noninteger order. For the
case when only one nonlinear term exists, the exact analytical solution of
the differential equation is determined as a cosine-Ateb function. Based on this solution, the asymptotic averaging procedure for solving the perturbed strong non-linear differential equation is developed. The method does not require the existence of the small parameter in the system. Special attention is given to the case when the dominant term is a linear one and to the case when it is of any non-linear order. Exact solutions of
the averaged differential equations of motion are obtained. The obtained results
are compared with “exact” numerical solutions and previously obtained
analytical approximate ones. Advantages and disadvantages of the suggested
procedure are discussed.
TL;DR: By using the subsidiary ordinary differential equation method, many explicit Jacobian elliptic periodic solutions of the cubic-quintic nonlinear optical transmission equation with higher-order dispersion nonlinear terms and self-steepening term are obtained as mentioned in this paper.
Abstract: By using the subsidiary ordinary differential equation method, many explicit Jacobian elliptic periodic solutions of the cubic-quintic nonlinear optical transmission equation with higher-order dispersion nonlinear terms and self-steepening term are obtained. The results are discussed.
TL;DR: In this paper, the structure of the singular set for a C1 smooth surface in the 3-dimensional Heisenberg group ℍ1 was studied and a Codazzi-like equation for the p-area element along the characteristic curves on the surface was discovered.
Abstract: In this paper, we study the structure of the singular set for a C1 smooth surface in the 3-dimensional Heisenberg group ℍ1. We discover a Codazzi-like equation for the p-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation helps us to analyze the local configuration of the singular set and the characteristic curves. In particular, we can estimate the size and obtain the regularity of the singular set. We understand the global structure of the singular set through a Hopf-type index theorem. We also justify the Codazzi-like equation by proving a fundamental theorem for local surfaces in ℍ1.
TL;DR: In this paper, the simplest equation method is used to construct exact solutions of nonlinear Schrodinger's equation and perturbed nonsmooth nonlinear Schröter's equation with Kerr law nonlinearity.
Journal Article•
TL;DR: In this article, the Bagley-Torvik equation is solved using an ordinary fractional differential equation, where the solution procedure is easier, more effective and straightforward than the traditional solution procedure, and the validity and the accuracy of this method is shown by the obtained results.
Abstract: In this paper we solve the Bagley-Torvik equation, which is an ordinary fractional differential equation , where the solution procedure is easier, more effective and straightforward. The validity and the accuracy of this method is shown by the obtained results.
TL;DR: A nonlinear Schrodinger equation with variable coefficients for surface waves on a large-scale steady nonuniform current has been derived without the assumption of a relative smallness of the velocity of the current.
Abstract: A nonlinear Schrodinger equation with variable coefficients for surface waves on a large-scale steady nonuniform current has been derived without the assumption of a relative smallness of the velocity of the current. This equation can describe with good accuracy the loss of modulation stability of a wave coming to a counter current, leading to the formation of so-called rogue waves. Some theoretical estimates are compared to the numerical simulation with the exact equations for a two-dimensional potential motion of an ideal fluid with a free boundary over a nonuniform bottom at a nonzero average horizontal velocity.
TL;DR: Numerical methods are proposed for solving the one-dimensional fractional Schrödinger differential equation with Dirichlet condition in the space variable.
Abstract: The first and second orders of accuracy difference schemes for the mixed problem for the multidimensional fractional Schrodinger differential equation with dependent coefficients are considered. Stability estimates for solutions of these difference schemes are obtained. Numerical methods are proposed for solving the one-dimensional fractional Schrodinger differential equation with Dirichlet condition in the space variable.
13 Aug 2012
TL;DR: In this paper, the mixed problem for a differential equation with involution is considered, and the authors propose a mixed solution for the problem of solving the problem with and without involution.
Abstract: In present paper, the mixed problem for a differential equation with involution is considered.
TL;DR: The repeated homogeneous balance method is used to construct exact traveling wave solutions of the (2 + 1) dimensional Boussinesq equation, in which the homogeneousbalance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation.
TL;DR: A necessary and sufficient condition for the reachable set, i.e., the set of states reachable from a ball ofInitial states at some time, of an ordinary differential equation to be convex is presented and convexity is guaranteed if the ball of initial states is sufficiently small.
Abstract: We present a necessary and sufficient condition for the reachable set, i.e., the set of states reachable from a ball of initial states at some time, of an ordinary differential equation to be convex. In particular, convexity is guaranteed if the ball of initial states is sufficiently small, and we provide an upper bound on the radius of that ball, which can be directly obtained from the right hand side of the differential equation. In finite dimensions, our results cover the case of ellipsoids of initial states. A potential application of our results is inner and outer polyhedral approximation of reachable sets, which becomes extremely simple and almost universally applicable if these sets are known to be convex. We demonstrate by means of an example that the balls of initial states for which the latter property follows from our results are large enough to be used in actual computations.
TL;DR: In this article, the authors give three new solutions to solve the 2D sine-Gordon equation with domain wall collision. But these solutions have not been presented in the literature.
Abstract: This paper gives three new solutions to solve the 2D sine-Gordon equation. Of particular interest is the Domain wall collision to 2D sine-Gordon equation which to the authors knowledge have not been presented in the literature.
Posted Content•
TL;DR: In this article, the potential flow of two-dimensional ideal incompressible fluid with a free surface is studied using the theory of conformal mappings and Hamiltonian formalism.
Abstract: The potential flow of two-dimensional ideal incompressible fluid with a free surface is studied. Using the theory of conformal mappings and Hamiltonian formalism allows us to derive exact equations of surface evolution. Simple form of the equations helped to discover new integrals of motion. These integrals are connected with the analytical properties of conformal mapping and complex velocity. Simple form of the equations also makes the numerical simulations of the free surface evolution very straightforward. In the limit of almost flat surface the equations can be reduced to the Hopf equation.
TL;DR: In this article, the complete Lie symmetry group classification of the dynamic fourth-order Euler-Bernoulli partial differential equation, where the elastic modulus, the area moment of inertia are constants and the applied load is a nonlinear function, was obtained.
Abstract: We obtain the complete Lie symmetry group classification of the dynamic fourth-order Euler-Bernoulli partial differential equation, where the elastic modulus, the area moment of inertia are constants and the applied load is a nonlinear function. In the Lie analysis, the principal Lie algebra which is two-dimensional extends in three cases, viz., the linear, the exponential, and the general power law. For each of the nontrivial cases, we determine symmetry reductions to ordinary differential equations which are of order four. In only one case related to the power law we are able to have a compatible initial-boundary value problem for a clamped end and a free beam. For these cases we deduce the corresponding fourth-order ordinary differential equations with appropriate boundary conditions. We provide an asymptotic solution for the reduced fourth-order ordinary differential equation corresponding to a clamped or free beam.