How to solve the polynomial ordinary differential equations

This paper discussed how to solve the polynomial ordinary differential equations. At first, we construct the theory of the linear equations about the unknown one variable functions with constant coefficients. Secondly, we use this theory to convert the polynomial ordinary differential equations into the simultaneous first order linear ordinary differential equations with constant coefficients and quadratic equations. Thirdly, we work out the general solution of the polynomial ordinary differential equations which is no longer concerned with the differential. Finally, we discuss the necessary and sufficient condition of the existence of the solution.     

Asymptotic expansion for the solution of singularly perturbed delay differential equations

Singular perturbation problems occur in many areas, including biochemical kinetics, genetics, plasma physics, and mechanical and electrical systems. For practical problems, one seeks a uniformly valid, readily interpretable approximation to a solution that does not behave uniformly. In this paper we extend singular perturbation theory in ordinary differential equations to delay differential equations with a fixed lag. We aim to give an explicit sufficient condition so that the solution of a class of singularly perturbed delay differential equations can be asymptotically expanded. O'Malley-Hoppensteadt technique is adopted in the construction of approximate solutions for such problems. Some particular phenomena different from singularly perturbed ordinary differential equations are discovered.     

Symmetries and Large Time Asymptotics of Compressible Euler Flows with Damping

Large time asymptotics of compressible Euler equations for a polytropic gas with and without the porous media equation are constructed in which the Barenblatt solution is embedded. Invariance analysis for these governing equations are carried out using the classical and the direct methods. A new second order nonlinear partial differential equation is derived and is shown to reduce to an Euler–Painlevé equation. A regular perturbation solution of a reduced ordinary differential equation is determined. And an exact closed form solution of a system of ordinary differential equations is derived using the invariance analysis.     

常微分方程中的反问题

研究某函数或函数组是什么常微分方程的通解或特解,这可以称为常微分方程中的反问题.这类问题,可以用"微分法"来解决.研究这类问题的意义在于通过利用"微分法"及"逆向思维方法"解决反问题的过程来加强对常微分方程理论内涵的深刻理解.     

On two classes of autonomous third-order nonlinear ordinary differential equations

Two autonomous, nonlinear, third-order ordinary differential equations whose dynamics can be represented by second-order nonlinear ordinary differential equations for the first-order derivative of the solution are studied analytically and numerically. The analytical study includes both the obtention of closed-form solutions and the use of an artificial parameter method that provides approximations to both the solution and the frequency of oscillations. It is shown that both the analytical solution and the accuracy of the artificial parameter method depend greatly on the sign of the nonlinearities and the initial value of the first-order derivative.     

Method of lines solutions of the parabolic inverse problem with an overspecification at a point

The present work is motivated by the desire to obtain numerical solution to a quasilinear parabolic inverse problem. The solution is presented by means of the method of lines. Method of lines is an alternative computational approach which involves making an approximation to the space derivatives and reducing the problem to a system of ordinary differential equations in the variable time, then a proper initial value problem solver can be used to solve this ordinary differential equations system. Some numerical examples and also comparison with finite difference methods will be investigated to confirm the efficiency of this procedure.     

Numerical diagnostics of solution blowup in differential equations

New simple and robust methods have been proposed for detecting poles, logarithmic poles, and mixed-type singularities in systems of ordinary differential equations. The methods produce characteristics of these singularities with a posteriori asymptotically precise error estimates. This approach is applicable to an arbitrary parametrization of integral curves, including the arc length parametrization, which is optimal for stiff and ill-conditioned problems. The method can be used to detect solution blowup for a broad class of important nonlinear partial differential equations, since they can be reduced to huge-order systems of ordinary differential equations by applying the method of lines. The method is superior in robustness and simplicity to previously known methods.     

Finite-difference methods for solving the reaction-diffusion equations of a simple isothermal chemical system

Numerical methods are proposed for the numerical solution of a system of reaction-diffusion equations, which model chemical wave propagation. The reaction terms in this system of partial differential equations contain nonlinear expressions. Nevertheless, it is seen that the numerical solution is obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic system, which is often required when integrating nonlinear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations, which model the well-stirred analogue of the chemical system. The first-order numerical methods proposed for the solution of this initialvalue problem are characterized to be implicit. However, in each case it is seen that the numerical solution is obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well-known, first-order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of the partial differential equations is seen to lead to two economical and reliable methods, one sequential and one parallel, for solving the travelling-wave problem. © 1994 John Wiley & Sons, Inc.     

Weak second order S-ROCK methods for Stratonovich stochastic differential equations

It is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit or semi-implicit methods with a wide variety of stability properties. However, for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis, such as those arising from stochastic partial differential equations, explicit methods with extended stability regions can be very effective. In the present paper our aim is to derive explicit Runge–Kutta schemes for non-commutative Stratonovich stochastic differential equations, which are of weak order two and which have large stability regions. This will be achieved by the use of a technique in Chebyshev methods for ordinary differential equations.     

The relationships between some types of partial differential equations and ordinary differential equations as well as their applications

In this article, we establish some relationships between several types of partial differential equations and ordinary differential equations. One application of these relationships is that we can get the exact values of the blowup time and the blowup rate of the solution to a partial differential equation by solving an ordinary differential equation. Another application of these relationships is that we can give the estimates for the spatial integration (or mean value) of the solution to a partial differential equation. We also obtain the lower and upper bounds for the blowup time of the solution to a parabolic equation with weighted function and space‐time integral in the nonlinear term.     

On the solution of boundary value problems with nonseparated multipoint and integral conditions

We suggest a numerical method for solving systems of linear nonautonomous ordinary differential equations with nonseparated multipoint and integral conditions. By using this method, which is based on the operation of convolution of integral conditions into local ones, one can reduce the solution of the original problem to the solution of a Cauchy problem for systems of ordinary differential equations and linear algebraic equations. We establish bounded linear growth of the error of the suggested numerical schemes. Numerical experiments were carried out for specially constructed test problems.     

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.     

Calculation of expansion coefficients of series in Chebyshev polynomials for a solution to a Cauchy problem

Two approaches are proposed to determine an initial approximation for the coefficients of an expansion of the solution to a Cauchy problem for ordinary differential equations in the form of series in shifted Chebyshev polynomials of the first kind. This approximation is used in an analytical method to solve ordinary differential equations using orthogonal expansions.     

Amplitude–shape approximation as an extension of separation of variables

Separation of variables is a well‐known technique for solving differential equations. However, it is seldom used in practical applications since it is impossible to carry out a separation of variables in most cases. In this paper, we propose the amplitude–shape approximation (ASA) which may be considered as an extension of the separation of variables method for ordinary differential equations. The main idea of the ASA is to write the solution as a product of an amplitude function and a shape function, both depending on time, and may be viewed as an incomplete separation of variables. In fact, it will be seen that such a separation exists naturally when the method of lines is used to solve certain classes of coupled partial differential equations. We derive new conditions which may be used to solve the shape equations directly and present a numerical algorithm for solving the resulting system of ordinary differential equations for the amplitude functions. Alternatively, we propose a numerical method, similar to the well‐established exponential time differencing method, for solving the shape equations. We consider stability conditions for the specific case corresponding to the explicit Euler method. We also consider a generalization of the method for solving systems of coupled partial differential equations. Finally, we consider the simple reaction diffusion equation and a numerical example from chemical kinetics to demonstrate the effectiveness of the method. The ASA results in far superior numerical results when the relative errors are compared to the separation of variables method. Furthermore, the method leads to a reduction in CPU time as compared to using the Rosenbrock semi‐implicit method for solving a stiff system of ordinary differential equations resulting from a method of lines solution of a coupled pair of partial differential equations. The present amplitude–shape method is a simplified version of previous ones due to the use of a linear approximation to the time dependence of the shape function. Copyright © 2007 John Wiley & Sons, Ltd.     

Ermakov–Pinney and Emden–Fowler Equations: New Solutions from Novel Bäcklund Transformations

We study the class of nonlinear ordinary differential equations y″ y = F(z, y²), where F is a smooth function. Various ordinary differential equations with a well-known importance for applications belong to this class of nonlinear ordinary differential equations. Indeed, the Emden–Fowler equation, the Ermakov–Pinney equation, and the generalized Ermakov equations are among them. We construct Bäcklund transformations and auto-Bäcklund transformations: starting from a trivial solution, these last transformations induce the construction of a ladder of new solutions admitted by the given differential equations. Notably, the highly nonlinear structure of this class of nonlinear ordinary differential equations implies that numerical methods are very difficult to apply.     

Transformation of an axialsymmetric disk problem for the helmholtz equation into an ordinary differential equation

The problem of solving the three-dimensional Helmholtz equation in the exterior of a circular disk is considered where radially symmetric Dirichlet data on the disk are assumed to be prescribed. This problem for example arises in the scattering of plane (sound) waves at an infinite plane screen with a circular aperture if the direction of the incident wave is normal to the screen, as well as in the process of diffusion through a circular hole. By applying the factorization technique developed in [N. GORENFLO, M. WERNER,Solution of a finite convolution equation with a Hankel kernel by matrix factorization, SIAM J. Math. Anal., 28 (1997), pp. 434–451] to the disk problem an equivalent ordinary differential equation is derived, whose solution leads directly to the solution of the disk problem. This differential equation belongs to a class of ordinary differential equations which are of higher complexity than the standard ordinary differential equations of mathematical physics. The examination of this new class of differential equations therefore is motivated.     

On the numerical solution of second order ordinary differential equations in the high-frequency regime

We describe an algorithm for the numerical solution of second order linear ordinary differential equations in the high-frequency regime. It is based on the recent observation that solutions of equations of this type can be accurately represented using nonoscillatory phase functions. Unlike standard solvers for ordinary differential equations, the running time of our algorithm is independent of the frequency of oscillation of the solutions. We illustrate this and other properties of the method with several numerical experiments.     

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTION FOR NONLINEAR SINGULAR 2mth-ORDER CONTINUOUS AND DISCRETE LIDSTONE BOUNDARY VALUE PROBLEMS

By fixed point theorem of a mixed monotone operators, we study Lidstone boundary value problems to nonlinear singular 2mth-order differential and difference equations, and provide sufficient conditions for the existence and uniqueness of positive solution to Lidstone boundary value problem for 2mth-order ordinary differential equations and 2mth-order difference equations. The nonlinear term in the differential and difference equation may be singular.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

Null space distributions—A new approach to finite convolution equations with a Hankel kernel

In some earlier publications it has been shown that the solutions of the boundary integral equations for some mixed boundary value problems for the Helmholtz equation permit integral representations in terms of solutions of associated complicated singular algebraic ordinary differential equations. The solutions of these differential equations, however, are required to be known on some infinite interval on the real line, which is unsatisfactory from a practical point of view. In this paper, for the example of one specific boundary integral equation, the relevant solutions of the associated differential equation are expressed by integrals which contain only one unknown generalized function, the support of this generalized function is no longer unbounded but a compact subset of the real line. This generalized function is a distributional solution of the homogeneous boundary integral equation. By this null space distribution the boundary integral equation can be solved for arbitrary right-hand sides, this solution method can be considered of being analogous to the method of variation of parameters in the theory of ordinary differential equations. The nature of the singularities of the null space distribution is worked out and it is shown that the null space distribution itself can be expressed by solutions of the associated ordinary differential equation.