Variable Coefficient Ordinary Differential Equations Research Articles

Approximate Symmetry Reduction Approach: Infinite Series Reductions to the KdV-Burgers Equation

For weak dispersion and weak dissipation cases, the (1+1)-dimensional KdV-Burgers equation is investigated in terms of approximate symmetry reduction approach. The formal coherence of similarity reduction solutions and similarity reduction equations of different orders enables series reduction solutions. For the weak dissipation case, zero-order similarity solutions satisfy the Painlevé II, Painlevé I, and Jacobi elliptic function equations. For the weak dispersion case, zero-order similarity solutions are in the form of Kummer, Airy, and hyperbolic tangent functions. Higher-order similarity solutions can be obtained by solving linear variable coefficients ordinary differential equations.

Approximate direct reduction method: infinite series reductions to the perturbed mKdV equation

The approximate direct reduction method is applied to the perturbed mKdV equation with weak fourth order dispersion and weak dissipation. The similarity reduction solutions of different orders conform to formal coherence, accounting for infinite series reduction solutions to the original equation and general formulas of similarity reduction equations. Painlevé II type equations, hyperbolic secant and Jacobi elliptic function solutions are obtained for zero-order similarity reduction equations. Higher order similarity reduction equations are linear variable coefficient ordinary differential equations.

Approximate homotopy symmetry method: Homotopy series solutions to the sixth-order Boussinesq equation

An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation, which arises from fluid dynamics. We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders, educing the related homotopy series solutions. Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation. Higher order similarity solutions can be obtained by solving linear variable coefficients ordinary differential equations. The auxiliary parameter has an effect on the convergence of homotopy series solutions. Series solutions and similarity reduction equations from the approximate symmetry method can be retrieved from the approximate homotopy symmetry method.

A Parallel Algorithm To Solve Large Stiff ODE Systems On Grid Systems

This paper introduces a parallel algorithm to solve large stiff ODE systems on distributed clusters, with computing nodes geographically distant from each other. This algorithm is based on the waveform relaxation method coupled with a sequential solver for differential equations systems. With respect to the standard PVODE algorithm (Parallel Variable-coefficient Ordinary Differential Equations solver; Byrne, George, and Hindmars 1999), it drastically reduces the number of messages exchanged between nodes which makes it less sensitive to slow communications. Thus, it is a coarse-grained algorithm well suited for grid environments connected via high latency networks. In this paper, we present various experiments which compare the PVODE solver and our algorithm and which show the benefits brought by this work.

A transformed rational function method and exact solutions to the [formula omitted] dimensional Jimbo–Miwa equation

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 Bäcklund transformation.

Solitary Wave and Wave Front as Viewed From Curvature**The project supported by National Natural Science Foundation of China under Grant No. 40305006 and the Ministry of Science and Technology of China through Special Public Welfare Project under Grant No. 2002DIB20070

The solitary wave and wave front are two important behaviors of nonlinear evolution equations. Geometrically, solitary wave and wave front are all plane curve. In this paper, they can be represented in terms of curvature c(s), which varies with arc length s. For solitary wave when , then its curvature c(s) approaches zero, and when , the curvature c(s) reaches its maximum. For wave front, when , then its curvature c(s) approaches zero, and when , the curvature c(s) is still zero, but . That is, is a turning point. When c(s) is given, the variance at some point (x, y) in stream line with arc length s satisfies a 2-order linear variable-coefficient ordinary differential equation. From this equation, it can be determined qualitatively whether the given curvature is a solitary wave or wave front.

Setting up second-order variable coefficient differential equation problems with known general solution

A large class of second-order variable coefficient ordinary differential equations for which there is a known solution are developed. This class forms the basis of a large number of examples which can be used when exploring the theory governing the solutions of linear differential equations.

The explicit solutions of the Sturm-Liouville problem

The author has obtained the explicit solution of boundary problems and initial problems of the typical form of the second order linear variable coefficient ordinary differential equation: y″( χ)+ q( χ) y( χ) = f( χ), a < χ < b, where q( χ) and f( χ) are arbitrary functons belonging to C[ a, b]. The solution are expressed in integral series explicitly by q( χ), f( χ), and boundary conditions or initial conditions. Besides, we prove the well-known basic properties and conditions of solvability for boundary problems of the above ordinary differential equation by using the explicit expressions.

Computer‐extended series for a source/sink driven gas centrifuge

AbstractWe have reformulated the general problem of internal flow in a modern, high speed gas centrifuge with sources and sinks in such a way as to obtain new, simple, rigorous closed form analytical solutions. Both symmetric and antisymmetric drives lead us to an ordinary differential equation in place of the usual inhomogeneous Onsager partial differential equation. Owing to the difficulties of exactly solving this sixth order, inhomogeneous, variable coefficient ordinary differential equation we appeal to the power of perturbation theory and techniques. Two extreme parameter regimes are identified, the so‐called semi‐long bowl approximation and a new short bowl approximation. Only the former class of problems is treated here. The long bowl solution for axial drive is the correct leading order term, just as for pure thermal drive. New O(1) results are derived for radial, drag and heat drives in two dimensions. Then regular asymptotic, even ordered power series expansions for the flow field are carried out on the computer to O(ε4) using MACSYMA. These approximations are valid for values of ε near unity. In the spirit of Van Dyke, one can carry out this expansion process, in theory, to apparently arbitrary order for arbitrary but finite decay length ratio. Curiously, the flows induced by axial and radial forces are proportional for asymptotically large source scale heights, x*. Corresponding isotope separation integral parameters will be given in a companion paper.