Solving evolutionary-type differential equations and physical problems using the operator method

We present a general operator method based on the advanced technique of the inverse derivative operator for solving a wide range of problems described by some classes of differential equations. We construct and use inverse differential operators to solve several differential equations. We obtain operator identities involving an inverse derivative operator, integral transformations, and generalized forms of orthogonal polynomials and special functions. We present examples of using the operator method to construct solutions of equations containing linear and quadratic forms of a pair of operators satisfying Heisenberg-type relations and solutions of various modifications of partial differential equations of the Fourier heat conduction type, Fokker–Planck type, Black–Scholes type, etc. We demonstrate using the operator technique to solve several physical problems related to the charge motion in quantum mechanics, heat propagation, and the dynamics of the beams in accelerators.     

Normal forms and unfoldings of linear systems in eigenspaces of (anti)-automorphisms of order two

In this article we classify normal forms and unfoldings of linear maps in eigenspaces of (anti)-automorphisms of order two. Our main motivation is provided by applications to linear systems of ordinary differential equations, general and Hamiltonian, which have both time-preserving and time-reversing symmetries. However, the theory gives a uniform method to obtain normal forms and unfoldings for a wide variety of linear differential equations with additional structure. We give several examples and include a discussion of the phenomenon of orbit splitting. As a consequence of orbit splitting we observe passing and splitting of eigenvalues in unfoldings.     

Solving differential equations using modified Picard iteration

Many classes of differential equation are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and engineering and include non-linear as well as linear differential equations. Examples involving partial as well as ordinary differential equations are presented. The method is easy to implement on a computer and the solutions so obtained are essentially power series. With its conceptual clarity (differential equations are integrated directly), its uniform methodology (the overall approach is the same in all cases) and its straightforward computer implementation (the integration and iteration procedures require only standard commercial software), the modified Picard methods offer obvious benefits for the teaching of differential equations as well as presenting a basic but flexible tool-kit for the solution process itself.     

Improved pedagogy for linear differential equations by reconsidering how we measure the size of solutions

For over 50 years, the learning of teaching of a priori bounds on solutions to linear differential equations has involved a Euclidean approach to measuring the size of a solution. While the Euclidean approach to a priori bounds on solutions is somewhat manageable in the learning and teaching of the proofs involving second-order, linear problems with constant co-efficients, we believe it is not pedagogically optimal. Moreover, the Euclidean method becomes pedagogically unwieldy in the proofs involving higher-order cases. The purpose of this work is to propose a simpler pedagogical approach to establish a priori bounds on solutions by considering a different way of measuring the size of a solution to linear problems, which we refer to as the Uber size. The Uber form enables a simplification of pedagogy from the literature and the ideas are accessible to learners who have an understanding of the Fundamental Theorem of Calculus and the exponential function, both usually seen in a first course in calculus. We believe that this work will be of mathematical and pedagogical interest to those who are learning and teaching in the area of differential equations or in any of the numerous disciplines where linear differential equations are used.     

Nonlinear stochastic differential delay equations

The methods used by Adomian and his co-workers to solve linear and nonlinear stochastic differential equations will be demonstrated to be applicable to differential equations, deterministic or stochastic, involving delays (constant, time dependent, or random).     

Isogroups of differential ideals of vector-valued differential forms: Application to partial differential equations

An older geometric technique for the study of invariance groups of partial differential equations, originally proposed by one of the authors and F. B. Estabrook, is generalized and extended to problems involving exterior equations for vector-valued or Lie algebra-valued exterior differential forms. Use of the method is demonstrated in the study of the symmetry groups of the two-dimensional Dirac equation and the full Yang-Mills free-field equations in Minkowski spacetime.     

Symmetry breaking of systems of linear second-order ordinary differential equations with constant coefficients

We show that the structure of the Lie symmetry algebra of a system of n linear second-order ordinary differential equations with constant coefficients depends on at most n-1 parameters. The tools used are Jordan canonical forms and appropriate scaling transformations. We put our approach to test by presenting a simple proof of the fact that the dimension of the symmetry Lie algebra of a system of two linear second-order ordinary differential with constant coefficients is either 7, 8 or 15. Also, we establish for the first time that the dimension of the symmetry Lie algebra of a system of three linear second-order ordinary differential equations with constant coefficients is 10, 12, 13 or 24.     

A nonlinear differential delay equation

A procedure reported elsewhere for solution of linear and nonlinear, deterministic or stochastic, delay differential equations developed by the authors as an extension of the first author's methods for nonlinear stochastic differential equations is now applied to a nonlinear delay-differential equation arising in population problems and studied by Kakutani and Markus. Examples involving time-dependent constants and even stochastic coefficients and delays can also be done.     

Apéry limits and special values of L-functions

We describe a general method to determine the Apéry limits of a differential equation that has a modular-function origin. As a by-product of our analysis, we discover a family of identities involving the special values of L-functions associated with modular forms. The proof of these identities is independent of differential equations and Apéry limits.     

On quintic Eisenstein series and points of order five of the Weierstrass elliptic functions

We employ a new constructive approach to study modular forms of level five by evaluating the Weierstrass elliptic functions at points of order five on the period parallelogram. A significant tool in our analysis is a nonlinear system of coupled differential equations analogous to Ramanujan??s differential system for the Eisenstein series on SL(2,?). The resulting relations of level five may be written as a coupled system of differential equations for quintic Eisenstein series. Some interesting combinatorial and analytic consequences result, including an alternative proof of a famous identity of Ramanujan involving the Rogers?CRamanujan continued fraction.     

Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

In this paper, we introduce new solutions for fuzzy differential equations as mixed solutions, and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro-differential operators of Volterra type. One example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions.     

Generalized normal forms for two-dimensional systems of ordinary differential equations with linear and quadratic unperturbed parts

Various definitions of normal forms for systems of ordinary differential equations are discussed. The notion of a generalized normal form and the problem of formal equivalency of systems of differential equations in terms of resonant equations are considered. The method of resonant equations is applied to two-dimensional systems whose unperturbed parts are linear in the first equation and quadratic in the second one.     

Differential quadrature solution of heat- and mass-transfer equations

The heat- and mass-transfer equations have an important role in various thermal and diffusion processes. These equations are nonlinear, due to the solution dependent diffusion coefficient and the source term. In this study, one- and two-dimensional nonlinear heat- and mass-transfer equations are solved numerically. To this end, the differential quadrature method is used to discretize the problem spatially and the resulting nonlinear system of ordinary differential equations in time are solved using the Runge–Kutta method. The solution is improved in time iteratively by solving considerably small sized linear system of resulting equations. To demonstrate its usefulness and accuracy, the proposed method is applied to four test problems, involving different nonlinearities.     

Hamiltonian and Variational Linear Distributed Systems

We use the formalism of bilinear- and quadratic differential forms in order to study Hamiltonian and variational linear distributed systems. It was shown in [1] that a system described by ordinary linear constant-coefficient differential equations is Hamiltonian if and only if it is variational. In this paper we extend this result to systems described by linear, constant-coefficient partial differential equations. It is shown that any variational system is Hamiltonian, and that any scalar Hamiltonian system is contained (in general, properly) in a particular variational system.     

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.     

On the existence of a lyapunov function as a quadratic form for impulsive systems of linear differential equations

A system of linear differential equations with pulse action at fixed times is considered. We obtain sufficient conditions for the existence of a positive-definite quadratic form whose derivative along the solutions of differential equations and whose variation at the points of pulse action are negative-definite quadratic forms regardless of the times of pulse action.     

On the Number of Zeros of Nonoscillatory Solutions to Higher-Order Linear Ordinary Differential Equations

 This paper is concerned with the problem of counting the number of zeros of bounded nonoscillatory solutions of higher-order linear ordinary differential equations involving a parameter. It is shown that a recent result for the second-order case is still valid for the higher-order case.     

初等函数Taylor级数的向量形式(英文)

推广了初等函数 Taylor级数的向量形式的一些结果 ,所考虑的初等函数 Taylor级数的向量形式涉及了三个复向量 .给出了在二阶常微分方程初值问题中的一个应用 .     

A general method for writing the dynamical equations of nonholonomic systems with ideal constraints

The basic notions of the dynamics of nonholonomic systems are revisited in order to give a general and simple method for writing the dynamical equations for linear as well as non-linear kinematical constraints. The method is based on the representation of the constraints by parametric equations, which are interpreted as dynamical equations, and leads to first-order differential equations in normal form, involving the Lagrangian coordinates and auxiliary variables (the use of Lagrangian multipliers is avoided). Various examples are illustrated.      

Contact linearization of nondegenerate Monge-Ampère equations

The present paper is devoted to the problem of transforming the classical Monge-Ampère equations to the linear equations by change of variables. The class of Monge-Ampère equations is distinguished from the variety of second-order partial differential equations by the property that this class is closed under contact transformations. This fact was known already to Sophus Lie who studied the Monge-Ampère equations using methods of contact geometry. Therefore it is natural to consider the classification problems for the Monge-Ampère equations with respect to the pseudogroup of contact transformations. In the present paper we give the complete solution to the problem of linearization of regular elliptic and hyperbolic Monge-Ampère equations with respect to contact transformations. In order to solve this problem, we construct invariants of the Monge-Ampère equations and the Laplace differential forms, which involve the classical Laplace invariants as coefficients.