Solution of Differential Equations by Using Legendre Wavelet Operational Matrix of Integration

Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration

Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration

This paper presents solutions of second-order ordinary differential equations on bond problems to estimate crack widths of reinforced concrete (RC) members subjected to repeated cyclic loads. Up to now, bond behaviors under repeated cyclic loads have usually been calculated by a difference scheme. In contrast, this research adopts analytical solutions of the differential equations to calculate bond behaviors in finite element models where a number of concrete cracks occur. The conventional differential equation for the bond problem has been available only under monotonic loading conditions. This paper extends the equation to cases where the reinforcement becomes plastic and the bond is subject to unloading/reloading conditions. The bond stress-slip relationship is modeled by a multilinear model that is composed of an envelope made of five lines, unloading/reloading paths, and a negative friction path. The solution of the differential equation is classified into 25 cases when the bond stress-slip relationship is on the envelope and seven cases when on the unloading/reloading paths or the negative friction path. The solutions are implemented into the smeared crack-based finite-element method (FEM) to evaluate the crack widths of a RC beam subject to repeated cyclic flexural/shear loads. The FEM program is incorporated with the discrete-like crack procedure developed by the authors and calculates stress redistributions due to cracking using the solutions of the differential equations. Typical distributions of reinforcement stress, bond stress, and slip along the reinforcement are presented.

In this paper, we study the growth of solutions of the second order linear complex differential equations insuring that any nontrivial solutions are of infinite order. It is assumed that the coefficients satisfy the extremal condition for Yang’s inequality and the extremal condition for Denjoy’s conjecture. The other condition is that one of the coefficients itself is a solution of the differential equation .

Seven common errors in finding exact solutions of nonlinear differential equations

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

In this review article, we study the recent investigations on the forms of solutions of systems difference equations and difference equations in terms of well-known integer sequences such as Fibonacci numbers, Padovan numbers. We focus on the papers given some interesting relationships both between the exact solutions of difference equations and the integer sequences and between the equilibrium points of difference equations and the golden ratio.

The methods in the previous sections provide asymptotic approximations for functions defined by integrals. Many of these functions are solutions of specific differential equations: the Bessel function is an example. In the case of functions defined as the solutions of differential equations which cannot be solved explicitly, these integral methods described above naturally cannot be used. (We group in the class of explicit solutions those given in the form of an integral.) In these circumstances we must resort to differential equation methods to find asymptotic approximations for the functions. The subject of asymptotic methods for solving differential equations is large. In this and the following sections we give a brief introduction to the subject. In the case of ordinary differential equations the book by Wasow (1965) discusses several aspects with detail and rigour. A large part of each of the books by Erdelyi (1956) and Jeffreys (1966) is devoted to ordinary differential equations. In the partial differential equation area the books by Van Dyke (1964) and Cole (1968)† are of importance in the particular area of asymptotic analysis called singular perturbation theory.KeywordsAsymptotic ExpansionAsymptotic SolutionAsymptotic FormLiouville EquationDominant TermThese 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.

In determining molecular weights with the ultracentrifuge by the equilibrium method, it is necessary to drive the rotor at relatively high speeds for very long periods of time. It would be a real advantage to be able to deduce molecular weights from the distribution of concentration within a cell placed in a rotor without waiting for equilibrium. The solution of a certain differential equation gives the theoretical distribution within the cell at any time, and a comparison of the experimentally determined concentrations with those deduced from theory should yield molecular weights without waiting for equilibrium. The exact solution of this differential equation is known, but it is not suitable for numerical work. The present paper presents an approximate solution from which numerical results can be obtained with ease.

For the purpose of this chapter, integrable systems are special kinds of non-linear differential equations (although one should note that there are also integrable difference equations, and the study of these is increasingly prominent). For essentially any system of differential equations, one has local existence theorems, so existence of solutions is not the issue. The question, rather, is what the solutions are like. For a non-linear differential equation chosen at random, one has no hope of writing down explicit solutions in terms of known functions. The classification of a function as ‘known’ may seem artificial, having more to do with what we’ve bothered to learn than the function itself; but in fact there is more to it than that. The known functions (e.g. trigonometric functions or elliptic functions) are known because they have nice properties, and can be defined using only a finite amount of information. The functions involved in the solutions of a generic nonlinear differential equation are so awful as to be literally indescribable in any but a tautological way (that is, one can really only define them as solutions of the differential equation). An extreme example of this is chaotic systems.

In practice the Galerkin method for solving elliptic partial differential equations yields equations involving certain integrals which cannot be evaluated analytically. Instead these integrals are approximated numerically and the resulting equations are solved to give “quadrature-Galerkin approximations” to the solution of the differential equation. Using a technique of J. Nitsche, L 2 {L^2} a priori error bounds are obtained for the difference between the solution of the differential equation and a class of quadrature-Galerkin approximations.

The first-passage time of a Markov process to a moving barrier is considered as a first-exit time for a vector whose components include the process and the barrier. Thus when the barrier is itself a solution of a differential equation, the theory of first-exit times for multidimensional processes may be used to obtain differential equations for the moments and density of the first-passage time of the process to the barrier. The procedure is first illustrated for first-passage-time problems where the solutions are known. The mean first-passage time of an Ornstein–Uhlenbeck process to an exponentially decaying barrier is then found by numerical solution of a partial differential equation. Extensions of the method to problems involving Markov processes with discontinuous sample paths and to cases where the process is confined between two moving barriers are also discussed.

The article addresses second-order algebraic differential equations that have a separated linear part and admit a finite-order integer function as a solution. All possible integer solutions of such equations are described. It is shown that all solutions are the solutions of certain second-order linear differential equations the coefficients of which are represented by rational functions. It has been demonstrated that any such integer function y = f(z) is either a solution of the algebraic equation R(z, exp{Q(z)}, y) ≡ 0 (where R is a polynomial of three variables, and Q(z) is a polynomial of one variable), or a solution of a differential equation with separable variables y′ = a(z)y (for some rational function a(z)).

It is well known that in many cases the solutions of a linear differential equation can be expressed as definite integrals, different solutions of the same equation being represented by integrals which have the same integrand, but different paths of integration. Thus, the various solutions of the hypergeometric differential equationcan be represented by integrals of the typethe path of integration being (for one particular solution) a closed circuit encircling the point t = 0 in the positive direction, then the point t = 1 in the positive direction, then the point t = 0 in the negative direction, and lastly the point t = 1 in the negative direction; or (for another particular solution) an arc in the t-plane joining the points t = 1 and t = ∞.

The existence of slowly and regularly varying solutions in the sense of Karamata implying nonoscillation is proved for a class of second order nonlinear retarded functional differential equations of Thomas-Fermi type. A motivation for such study is the extensively developed theory offering a number of properties of regularly and slowly varying functions ([2]) - consequently of such solutions of differential equations. As an illustration, the precise asymptotic behaviour for $t\rightarrow \infty$ of the slowly varying solutions for a subclass of considered equations is presented.