First-order Integro-differential Equations Research Articles

New results for the periodic boundary value problem for impulsive integro-differential equations

This paper discusses the periodic boundary value problem for first-order nonlinear impulsive integro-differential equations. We prove some new comparison principles, and then establish new existence results for extremal solutions by using these principles and the monotone iterative technique. These results improve and extend all known relevant conclusions from the literature. Some examples are also given to illustrate the advantage of our results.

Second-Order Radiative Transfer Equation and Its Properties of Numerical Solution Using the Finite-Element Method

The original radiative transfer equation is a first-order integrodifferential equation, which can be taken as a convection-dominated equation. The presence of the convection term may cause nonphysical oscillation of solutions. This type of instability can occur in many numerical methods, including the finite-difference method and the finite-element method, if no special stability treatment is used. To overcome this problem, a second-order radiative transfer equation is derived, which is a diffusion-type equation similar to the heat conduction equation for an anisotropic medium. The consistency of the second-order radiative transfer equation with the original radiative transfer equation is demonstrated. The perturbation characteristics of error are analyzed and compared for both the first- and second-order equations. Good numerical properties are found for the second-order radiative transfer equation. To show the properties of the numerical solution, the standard Galerkin finite-element method is employed to solve the second-order radiative transfer equation. Four test problems are taken as examples to check the numerical properties of the second-order radiative transfer equation. The results show that the standard Galerkin finite-element solution of the second-order radiative transfer equation is numerically stable, efficient, and accurate.

Extremal Solutions of Periodic Boundary Value Problems for First-Order Impulsive Integrodifferential Equations of Mixed-Type on Time Scales

We consider the existence of minimal and maximal solutions of periodic boundary value problems for first-order impulsive integrodifferential equations of mixed-type on time scales by establishing a comparison result and using the monotone iterative technique.

Self-similar solutions of the magnetohydrodynamic boundary layer system for a dilatable fluid

A rigorous mathematical analysis is given for a magnetohydrodynamic boundary layer problem, which arises in the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting dilatable fluid along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. In the self-similar case, the problem is transformed into a third-order nonlinear ordinary differential equation with certain boundary conditions, which is proved to be equivalent to a singular initial value problem for an integro-differential equation of first order. With the aid of the singular initial value problem, the uniqueness, existence, and nonexistence results for generalized normal solutions are established.

On the similarity solutions of magnetohydrodynamic flows of power-law fluids over a stretching sheet

A rigorous mathematical analysis is given for a magnetohydrodynamics boundary layer problem, which arises in the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluid along a stretching flat sheet in the presence of an exterior magnetic field orthogonal to the flow. In the self-similar case, the problem is transformed into a third-order nonlinear ordinary differential equation with certain boundary conditions, which is proved to be equivalent to a singular initial value problem for an integro-differential equation of first order. With the aid of the singular initial value problem, the uniqueness and existence results for (generalized) normal solutions are established and some properties of these solutions are explored.

Nonlinear asymptotic analysis in elastica of straight bars—analytical parametric solutions

It is shown that by a series of admissible functional transformations the already derived (third-order) strongly nonlinear ordinary differential equation (ODE), describing the elastica buckling analysis of a straight bar under its own weight [Int.J.Solids Struct.24(12), 1179–1192, 1988, The Theory of Elastic Stability, McGraw-Hill, New York, 1961], is reduced to a first-order nonlinear integrodifferential equation. The absence of exact analytic solutions of the reduced equation leads to the conclusion that there are no exact analytic solutions in terms of known (tabulated) functions of this elastica buckling problem. In the limits of large or small values of the slope of the deflected elastica, we expand asymptotically the above integrodifferential equation to nonlinear ODEs of the Emden–Fowler or Abel nonlinear type. In these cases, using the solution methodology recently developed in Panayotounakos [Appl. Math. Lett. 18:155–162, 2005] and Panayotounakos and Kravvaritis [Nonlin. Anal. Real World Appl., 7(2):634–650, 2006], we construct exact implicit analytic solutions in parametric form of these types of equations and thus approximate implicit analytic solutions of the original elastica buckling nonlinear ODE.

New comparison results for impulsive integro-differential equations and applications

We prove some new maximum principles for ordinary integro-differential equations. This allows us to introduce a new definition of lower and upper solutions which leads to the development of the monotone iterative technique for a periodic boundary value problem related to a nonlinear first-order impulsive integro-differential equation.

Nonlinear boundary problem of first order impulsive integro-differential equations

This paper focuses on first-order impulsive integro-differential equations with nonlinear boundary. The lower and upper solutions combined with monotone iterative techniques are used to obtain the existence of extremal solutions of the problem. Numerical examples are given to prove its validity.

Asymptotic solutions to the non-linear cantilever elastica

In this work it is shown that by a series of admissible functional transformations the constructed higher-order strongly non-linear differential equation (ODE), describing the elastica of a cantilever due to a terminal generalized concentrated, as well as to a lateral uniformly distributed loading, is reduced to a first-order non-linear integrodifferential equation consisting of the first intermediate integral of the original equation. The absence of exact analytic solutions in terms of known (tabulated) functions of the above reduced equation leads to the conclusion that there are no exact analytic solutions of this complicated elastica problem. In the limits of small values of the slope parameter of the deflected elastica, we expand asymptotically the above integrodifferential equation to non-linear ODEs of the generalized Emden–Fowler types, exact analytic solutions of which are constructed in parametric form.

Initial value problem for first-order integro-differential equation of Volterra type on time scales☆

Initial value problem for first-order integro-differential equation of Volterra type on time scales☆

Initial value problem for first-order integro-differential equation of Volterra type on time scales

In this paper, we combine the monotone iterative technique with fixed point theory to investigate the extremal solutions of nonlinear integro-differential equations on time scales. Several important lemmas and existence criteria are obtained.

On the reduction and the existence of approximate analytic solutions of some basic nonlinear ODEs in mathematical physics and nonlinear mechanics

Using series of admissible functional transformations we reduce the one-dimensional axisymmetric nonlinear Schrödinger (NLS) equation, as well as the forced damped nonlinear Duffing (NLD) equation to equivalent nonlinear first-order integrodifferential equations. The forced undamped (NLD) equation results as a special case. The reduced integrodifferential equations are exact. In the limits of small or large values of the parameters characterizing these nonlinear problems, we prove that further reductions lead to first-order nonlinear ordinary differential equations which, except in case of the (NLS) equation, are of the Abel classes. The approximate reduced (NLS) equation admits exact analytic solutions. On the other hand, taking into account the known exact analytic solutions of the equivalent Abel classes of equations we show that there do not exist analytic solutions of the above two nonlinear Duffing oscillators. However, if further asymptotic approximations take place, new approximate analytic solutions concerning the (NLD) equations are constructed.

Remarks on some series expansions associated with certain products of the incomplete Gamma functions

In several recent works, some interesting generalizations of the first-order Volterra-type integro-differential equation governing the unsaturated behavior of the free electron laser (FEL) were introduced and investigated by making use of fractional calculus (that is, calculus of integrals and derivatives of an arbitrary real or complex order). Among other things, it is observed here that an expansion formula for the confluent hypergeometric function in a series of the product of two entire (integral) incomplete Gamma functions does not hold true as asserted and applied in these earlier works. Some necessary corrections and possible remedies are also pointed out.

On the reduction of some second-order nonlinear ODEs in physics and mechanics to first-order nonlinear integrodifferential and Abel’s classes of equations

Second-order ordinary differential equations (ODEs) with strong nonlinear stiffness terms (cubic nonlinearities) governing wave motions, dynamic crack propagations, nonlinear oscillations etc. in physics and nonlinear mechanics are analyzed. Selecting as guide line a second-order nonlinear ODE of the form of the forced Duffing equation and using admissible functional transformations it is possible to reduce it to an equivalent first-order nonlinear integrodifferential equation. The reduced equation is exact. In the limits of small or large values of the parameter characterizing this nonlinear problem, it is shown that further reductions lead to a nonlinear ODE of the Abel classes. Taking into account the known exact analytic solutions of this equivalent equation it is proved that there does not exist an exact analytic solution of this type of equations. However, in cases when convenient functional relations connecting all parameters of the corresponding null equation and the characteristics of the driving force exist, approximate analytic solutions to the problem under consideration are provided.

Recovering a leading coefficient and a memory kernel in first-order integro-differential operator equations

We are concerned with the identification of the scalar functions a and k in the convolution first-order integro-differential equation u′(t)−a(t)Au(t)−k∗Bu(t)=f(t), 0⩽ t⩽ T, k∗v(t)=∫ 0 tk(t−s)v(s) ds , in a Banach space X, where A and B are linear closed operators in X, A being the generator of an analytic semigroup of linear bounded operators. Taking advantage of two pieces of additional information, we can recover, under suitable assumptions and locally in time, both the unknown functions a and k. The results so obtained are applied to an n-dimensional integro-differential identification problem in a bounded domain in R n .

Multiple positive solutions for first order nonlinear integro-differential equations in Banach spaces

By using the fixed point index theory, the author obtains the existence of two positive solutions for a boundary value problem of first-order nonlinear integro-differential equations on an infinite interval in a real Banach space.

Perturbation theory analysis of competition in a heterogeneous population

A physiologically structured population model of T cell populations at the end of an immune response is proposed and analysed. The proposed model describes a T cell population with a structured death rate mediated by cytokine receptor surface expression. This provides a structuring that determines the ability of a T cell to compete for survival cytokines. Homogeneous differentiation into memory combined with heterogeneous survival abilities leads to a system of nonlinear first-order integro differential equations. The mathematical model is studied analytically by a perturbation approach which agrees well with numerical integrations. We find that the structured death rate leads to selection of T cells expressing high cytokine receptor levels. Selection in this model is independent of the initial T cell distribution which allows us to define a universal selection curve, a property lost under receptor downregulation (i.e. loss of cytokine receptors from the surface). Furthermore, we examine the effects of a population dependent memory differentiation rate and of cytokine receptor downregulation on selection. A population dependent memory differentiation rate increases selection significantly while putting an upper bound on the number of T cells differentiating into memory. Under receptor downregulation a dramatic change of the memory T cell distribution is induced. We find that there exists a critical ratio of the receptor downregulation rate to the average T cell death rate for which selection becomes maximal. Our analysis reveals that this maximum property is the result of an interplay between single cell dynamics and population dynamics. We propose that the selection studied in this paper plays an important role in the selection of optimal T cells for the memory pool.

The numerical solution of first order delay integro-differential equations by spline functions

In this paper, a new method for approximating the solution of nonlinear first-order delay integro-differential equations is presented. Boundedness of the approximate solution, conver-gence results, as well as numerical examples are given.

Periodic boundary value problems for first-order integro-differential equations of mixed type

The existence of at least one solution and the existence of extreme solutions of periodic boundary value problems for first-order integro-differential equations of mixed type are studied, in the presence of generalized upper and lower solutions. The discussion is based on new comparison theorems and coincidence degree and monotone iterative methods.

Nuclear resonant forward scattering of x rays: Time and space picture

The problem of forward resonant scattering of x rays by an ensemble of nuclei is being solved directly in time and space. The wave equation describing the propagation of the radiation through the nuclear ensemble is derived. It is a first-order integrodifferential equation. Its kernel is a double time function $K(t,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t})$ that represents a coherent single scattering response of the nuclear system at time t to excitation at $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}.$ The explicit form of the kernel is defined by the character of interactions, the nuclei experience with the environment and by the character of their spatial motion. A general procedure of the solution of the wave equation is introduced that is independent of the type of the kernel. Examples for various kernels are presented and discussed for some particular cases: collective or diffusive motion of nuclei in space, thermal lattice vibrations, time-independent hyperfine interactions, and time-dependent hyperfine interactions due to atomic spin fluctuations or external magnetic-field switching.