NAGUMO CONDITIONS FOR ORDINARY DIFFERENTIAL EQUATIONS

The study of asymptotic representations of solutions of differential equations, in particular, of the second order differential equations, which contain nonlinearities of various types in the right-hand side, play an important role in the development of the qualitative theory of differential equations. This paper considers the type of differential equations of the second order, which contain in the right part the product of a regularly varying function from an unknown function and a rapidly varying function from the derivative of an unknown function when the corresponding arguments tend to zero or infinity. Necessary and sufficient conditions for the existence of slowly varying Pω(Y0, Y1, ±∞)-solutions of such equations have been obtained. Asymptotic representations of such solutions and their first-order derivatives are also obtained. Note that when additional conditions are imposed on the coefficients of the characteristic equation, such Pω(Y0, Y1, ±∞)-solutions of the equation exist as a one-parameter family. Similar results were obtained earlier when considering second-order equations, which contain in the right-hand side the product of a rapidly varying function from an unknown function and a regularly varying function from the derivative of an unknown function when the arguments tend to zero or infinity. For the equations considered in this paper, similar results are new.

Numerical solution for high order differential equations using a hybrid neural network—Optimization method

THIS investigation is primarily concerned with the solution of boundary-value problems by means of automatic computing equipment. Specifically, it is concerned with the solution of ordinary differential equations in which certain conditions (boundary values) are specified at two or more values of the independent variable. Such problems are generally called boundary-value problems; however, it must be emphasized that this report is restricted to the solution of ordinary differential equations, and not to the solution of partial differential equations.

In the preceding chapter, we developed rather completely the regularity properties of weak and strong solutions of certain first order variational problems and second order differential equations involving only one unknown function. Complete results concerning the solutions of the corresponding problems involving several unknown functions were obtained only in the case where v = 2; these results were obtained by the writer before the war and were described in Chapter 1. In the preceding chapter some first differentiability results were obtained for certain systems of equations but these results did not imply the continuity of the first derivatives. In 1952, the writer presented a paper (Morrey [10]) at the Arden House Conference on Partial Differential Equations, in which it was proved that any vector solution of class C1 of a regular variational problem of class C n μ , n ≥ 2, was also of class C n μ . These results still leave a gap in the theory for systems, which can only be filled by an extension of the De Giorgi-Nash results, developed in § 5–3, to systems or some entirely new device. But, in proving the C n gm differentiability results for systems of equations, the writer was forced to use the very important formulas of F. John ([1], [3]) for fundamental solutions and to use methods which are appropriate for the discussion of elliptic systems of higher order. Hence the inclusion of this chapter in this book.KeywordsWeak SolutionElliptic SystemGeneral BoundaryPrincipal PartRegularity TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction. This study investigates the possibility of increasing the accuracy of numerically solving boundary value problems using the modified Bubnov-Galerkin method with a linear ordinary differential equation, where the coefficients and the right-hand side are continuous functions. The order of the differential equation n must be less than the number of coordinate functions m.Materials and Methods. A modified Petrov-Galerkin method was used to numerically solve the boundary value problem. It employs a system of linearly independent power-type basis functions on the interval [−1,1], each normalized by the unit Chebyshev norm. The system of linear algebraic equations includes only the linearly independent boundary conditions of the original problem.Results. For the first time, an integral quadrature formula with a 22nd order error was developed for a uniform grid. This formula is used to calculate the matrix elements and coefficients in the right-hand side of the system of linear algebraic equations, taking into account the scalar product of two functions based on the new quadrature formula. The study proves a theorem on the existence and uniqueness of a solution for boundary value problems with general non-separated conditions, provided that n linearly independent particular solutions of a homogeneous differential equation of order n are known.Discussion and Conclusion. The hydrodynamic problem in a viscous strong boundary layer with a third-order equation was precisely solved. The analytical solution was compared with its numerical counterpart, and the uniform norm of their difference did not exceed 5·10‒15. The formulas derived using the generalized Bubnov-Galerkin method may be useful for solving boundary value problems with linear ordinary differential equations of the third and higher orders.

Semilocal approach to nonlocal equations

<abstract><p>Differential equations of second order with impulses at random moments are set up and investigated in this paper. The main characteristic of the studied equations is that the impulses occur at random moments which are exponentially distributed random variables. The presence of random variables in the ordinary differential equation leads to a total change of the behavior of the solution. It is not a function as in the case of deterministic equations, it is a stochastic process. It requires combining of the results in Theory of Differential Equations and Probability Theory. The initial value problem is set up in appropriate way. Sample path solutions are defined as a solutions of ordinary differential equations with determined fixed moments of impulses. P-moment generalized exponential stability is defined and some sufficient conditions for this type of stability are obtained. The study is based on the application of Lyapunov functions. The results are illustrated on examples.</p></abstract>

Solutions of First Order Differential Equations which are Solutions of Linear Differential Equations of Higher Order

The mathematical formulation of characterization problems leads often to an ordinary differential equation for the characteristic function f(t) of the population distribution function. This is the case when a population is characterized by the independence of two statistics or by a regression property involving two statistics. The solution of such characterization problems is carried out in three distinct steps. One has first to derive the differential equation which expresses the characterizing property in mathematical terms. The next step is the determination of the solutions of this differential equation. The third step requires the selection of those solutions which can be characteristic functions, i.e. which are non-negative definite. This final step is usually the most difficult part of the problem. In this connection it is of great interest to derive analytical properties of the non-negative definite solutions of certain differential equations. In this paper we survey some results of this kind and we show also how these results can help in the solution of characterization problems.

This study introduced real life application of first order differential equation. In this paper We basically discussed about different types of differential equation and the solution of first order differential equation and application of first order differential equation in different field of science and technology. Further, Newton’s law of cooling and orthogonal trajectory has been incorporated. Study about convective boundary condition and it is used for increasing the temperature.

In this paper we obtain existence conditions and an explicit closed form expression of the general solution of twopoint boundary value problems for coupled systems of second order differential equations with a singularity of the first kind. The approach is algebraic and is based on a matrix representation of the system as a second order Euler matrix differential equation that avoids the increase of the problem dimension derived from the standard reduction of the order method.

This paper investigates the existence of positive solutions of nonresonant singular boundary value problem of second order differential equations. A necessary and sufficient condition for the existence of C[0, 1] positive solutions as well as C1[0, 1] positive solutions is given by means of the method of lower and upper solutions with the fixed point theorems.

Comparison theorems for ordinary differential equations with general boundary conditions

In the present study, we have investigated the differential equations of order four to evolve the methods to achieve the solution for differential equations. Absolute Stability Region (ASR) of the differential equations has been examined. Numerical Differentiation (ND) and Differential Transform Method (DTM) which are suggested and derived in this article are much suitable to understand the solutions of differential equations of fourth order. Both the methods are applied to some differential equations, numerical examples and results are presented to outline the capability and robustness of our strategies and compared them with that of exact solution.

Differential equations of fractional order are widely used in physics, chemistry as well as engineering fields, this is the main reason that the approximate solution of fractional differential equations becomes a hot topic. In this paper, a numerical scheme for a class of fractional boundary value problems (FBVPs) is presented. In this approach, the FBVPs are expressed in terms of Caputo's fractional derivative. This scheme is based on exponential spline functions consisting of a polynomial part of degree one and an exponential part. For convergence analysis of this method, it is assumed that the exact solution of fractional boundary value problem belongs to a class of $$C^{6}$$C6-functions. Numerical examples are considered to illustrate the practical usefulness of this method and comparison show that this scheme is more accurate than the existing method Zahra and Elkholy (Numer Algorithms 59:373---391, 2012).