Finite Difference Schemes and Classical Transcendental Functions

The free transverse oscillations of the rod of uniform cross section taking into account its own weight are considered in this work. The appropriate partial differential equation of transverse oscillations of a rod was reduced to two ordinary differential equations for the time function and the amplitude function of deflections. Concurrent with the differential equations for the amplitude state, the equivalent system of differential equation is considered. In total, the exact solution of the initial partial differential equation of transverse oscillations of a rod, expressed in nondimensional fundamental functions and initial parameters is attained. The method of power series was used for the construction of fundamental functions. Due to the exact solution, the formulas in an explicit form for dynamic variables of the state of a rode – deflections, angular displacement, bending moment and transverse force – were defined. The analytical form for equation of free oscillation frequency is defined. That has limited the finding of frequency to definition the unknown non-dimensional parameter through the frequency equation. As a result, the presence of derivative exact solutions provides the possibility to investigate the free oscillations of rod with various types of boundary conditions.

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.

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.

New explicit and implicit “improved euler” methods for the integration of ordinary differential equations

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

Seven common errors in finding exact solutions of nonlinear differential equations

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

This work presents the application of the power series method (PSM) to find solutions of nonlinear delay differential equations of pantograph type (PDDEs). Three equations are solved to show that PSM can provide analytical solutions of PDDEs in convergent series form. The nonlinear pantograph cases study are: a first order equation, a second order equation, and a second order singular equation. Additionally, we present the post-treatment of the power series solutions with the Laplace-Pad ´ e (LP) resummation method as a powerful technique to find exact solutions. The proposed methodology possesses a simple procedure based on a few straightforward steps and it does not depend on a perturbation parameter.

Within the theory of linear systems, the differential equation associated with sinusoidal waves has a central importance in signal analysis. Free and forced oscillations or transient and steady-state responses can be understood as general and particular solutions of linear differential equations. Resonances may be formulated as solutions of the differential equation for a system with a single degree of freedom. Given observation records of a system, estimating source information is a fundamental issue in machine diagnostics. Taking an observation signal as a solution of a differential equation, the waveform of an external source acting on a system of a single degree of freedom can be recovered using the differential equation because in principle the transient response is independent of the source effect. A mixture of transient and steady-state solutions creates beats if the frequency of the source is close to the frequency of the resonance. The differential equation can be extended to the wave equation that governs wave propagation in a medium. A typical phenomenon of wave propagation is the standing wave, which is a superposition of direct and reflection waves, subject to there being no magnitude differences between the two waves. If the coefficient of reflection is given by a complex number, the phase difference moves the nodes; however, the standing waves disappear as the magnitude difference increases.

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.

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 .

The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations

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.