On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order

In the paper we study the existence of solutions of a nonlinear quadratic Volterra integral equation of fractional order. This equation is considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval. Moreover, we show that solutions of this integral equation are locally attractive.     

On solutions of a quadratic integral equation of Hammerstein type

We study the solvability of a nonlinear quadratic integral equation of Hammerstein type. Using the technique of measures of noncompactness we prove that this equation has solutions on an unbounded interval. Moreover, we also obtain an asymptotic characterization of these solutions. Several special cases of this integral equation are discussed and applications to real world problems are indicated.     

一类共形不变摄动积分方程正解的存在性

本文讨论了一类共形不变摄动积分方程正解的存在性. 我们证明了:当参数对(p, q) 属于集合(-n, 0) × (0,∞) 且pq + p + 2n = 0 时, 对应摄动积分方程存在正解; 而当参数对(p, q) 属于集合(0,∞)×(-∞, 0) 也满足pq +p+2n = 0 时, 摄动积分方程不存在非负解. 这与原共形不变积分方程有着本质的不同, 此结果隐含着这类积分方程正解的存在性取决于解在无穷远处的性态.     

Approximations of Solutions to Abstract Neutral Functional Differential Equation

In the present work, we study the approximations of solutions to the abstract neutral functional differential equations with bounded delay. We consider an associated integral equation and a sequence of approximate integral equations. We establish the existence and uniqueness of the solutions to every approximate integral equation using the fixed point arguments. We then prove the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Next, we consider the Faedo–Galerkin approximations of the solutions and prove some convergence results. Finally, we demonstrate the application of the results established.     

On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces

The main purpose of this paper is to deal with almost automorphic and asymptotically almost automorphic solutions of the initial value problem as well as the nonlinear Volterra integral equation in Banach spaces. We obtain a collection of existence results of such solutions to these equations. We investigate also a topological structure of such solution sets. Moreover, we prove Aronszajn-type theorems for solutions of the initial value problem as well as the nonlinear Volterra integral equation, defined on the whole real line.     

奇摄动Volterra型积分微分方程Robin问题

本文利用上、下解证明了Volterra型积分微分方程解的存在性。然后，应用所获得的微分不等式理论，在适当的假设下，通过构造特殊的上、下解函数，证明Volterra型 奇摄动积分微分方程解的存在性，并给出一致有效的解的渐近估计。     

Jacob’s ladders and the almost exact asymptotic representation of the Hardy-Littlewood integral

In this paper we introduce a nonlinear integral equation such that the system of global solutions to this equation represents the class of a very narrow beam as T → ∞ (an analog of the laser beam) and this sheaf of solutions leads to an almost-exact representation of the Hardy-Littlewood integral (1918). The accuracy of our result is essentially better than the accuracy of related results of Balasubramanian, Heath-Brown, and Ivic.     

On the solution of the Chapman-Kolmogorov integral equation in the form of a series

The Chapman-Kolmogorov nonlinear integral equation is of fundamental importance in the theory of Markov stochastic processes. The solution to this equation is the transition probability density. It is usually solved by means of reducing to a linear equation. In 1932, S.N. Bernshtein formulated the problem of whether this equation can be solved directly. In 1962, O.V. Sarmanov found such solutions in terms of a bilinear series for a stationary Markov process. In 2007, the author obtained several solutions in the form of integrals of the product of two kernels of known integral transforms. In this paper, without imposing Sarmanov’s constraints, we derive solutions in the form of a series whose terms contain the product of two orthogonal functions. The results are illustrated by examples in which the series converges to a simple function.     

Approximate solution of one singular integro-differential equation

In this paper we construct and theoretically justify a computational scheme for solving the Cauchy problem for a singular integro-differential equation of the first-order, where the integral over a segment of the real axis is understood in the sense of the Cauchy principal value. In addition, we study and solve approximately the integral equation with a special logarithmic kernel. We obtain uniform estimates for errors of approximate formulas. Orders of errors of approximate solutions are proved to be proportional to the order of the approximation error for the derivative of the density of the singular integral in the integro-differential equation.     

Traveling wave solutions to a reaction-diffusion equation

In this paper, we restrict our attention to traveling wave solutions of a reaction-diffusion equation. Firstly we apply the Divisor Theorem for two variables in the complex domain, which is based on the ring theory of commutative algebra, to find a quasi-polynomial first integral of an explicit form to an equivalent autonomous system. Then through this first integral, we reduce the reaction-diffusion equation to a first-order integrable ordinary differential equation, and a class of traveling wave solutions is obtained accordingly. Comparisons with the existing results in the literature are also provided, which indicates that some analytical results in the literature contain errors. We clarify the errors and instead give a refined result in a simple and straightforward manner.      

Faedo–Galerkin Approximation of Solution for a Nonlocal Neutral Fractional Differential Equation with Deviating Argument

In this work, we study a class of nonlocal neutral fractional differential equations with deviated argument in the separable Hilbert space. We obtain an associated integral equation and then, consider a sequence of approximate integral equations. We investigate the existence and uniqueness of the mild solution for every approximate integral equation by virtue of the theory of analytic semigroup theory via the technique of Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. The Faedo–Galerkin approximation of the solution is studied and demonstrated some convergence results. Finally, we give an example.

     

Traveling wave solutions to a reaction-diffusion equation

In this paper, we restrict our attention to traveling wave solutions of a reaction-diffusion equation. Firstly we apply the Divisor Theorem for two variables in the complex domain, which is based on the ring theory of commutative algebra, to find a quasi-polynomial first integral of an explicit form to an equivalent autonomous system. Then through this first integral, we reduce the reaction-diffusion equation to a first-order integrable ordinary differential equation, and a class of traveling wave solutions is obtained accordingly. Comparisons with the existing results in the literature are also provided, which indicates that some analytical results in the literature contain errors. We clarify the errors and instead give a refined result in a simple and straightforward manner.     

On a class of ill-posed non-linear integral equations arising in the interpretation of indirect measurements

In this paper we consider a class of specific Urysohn integral equations for which the solutions are only determined with the exception of rearrangements of function values and associated arguments. As an alternative to Tikhonov's regularization method approximating minimum-norm solutions for this ill-posed class of inverse problems, a constrained least-squares approach is presented. This approach is aimed at finding decreasing rearrangements serving as appropriate solution representatives. It is shown that the inverses of these decresing solutions solve a Fredholm linear integral equation of the first kind.     

非线性积分方程解的指数与解的个数

本文对型非线性积分方程,给出了一个由Fredholm行列式表示其解的指数的公式,并且讨论了利用Fredholm行列式判断其解的个数的方法。 引理1 设线性积分算子A:C(G)→C(G) (Au)(x)=integral from G (K(x,y)u(y)dy) (1) 其中GR~N为有界闭域,K(x,y)在G×G上连续。从而A全连续,相应于A的Fredholm行列式F-det[I-A]为     

A numerical method for solving the time fractional Schrödinger equation

In this article, we proposed a new numerical method to obtain the approximation solution for the time-fractional Schrödinger equation based on reproducing kernel theory and collocation method. In order to overcome the weak singularity of typical solutions, we apply the integral operator to both sides of differential equation and yield a integral equation. We divided the solution of this kind equation into two parts: imaginary part and real part, and then derived the approximate solutions of the two parts in the form of series with easily computable terms in the reproducing kernel space. New bases of reproducing kernel spaces are constructed and the existence of approximate solution is proved. Numerical examples are given to show the accuracy and effectiveness of our approach.     

Antiplane shear deformations in a linear theory of elasticity with microstructure

In this paper, we solve fundamental boundary value problems in a theory of antiplane elasticity which includes the effects of material microstructure. Using the real boundary integral equation method, we reduce the fundamental problems to systems of singular integral equations and construct exact solutions in the form of integral potentials.     

Euler integral symmetries for the confluent Heun equation and symmetries of the Painlevé equation PV

Euler integral symmetries relate solutions of ordinary linear differential equations and generate integral representations of the solutions in several cases or relations between solutions of constrained equations. These relations lead to the corresponding symmetries of the monodromy matrices for the differential equations. We discuss Euler symmetries in the case of the deformed confluent Heun equation, which is in turn related to the Painlevé equation PV. The existence of symmetries of the linear equations leads to the corresponding symmetries of the Painlevé equation of the Okamoto type. The choice of the system of linear equations that reduces to the deformed confluent Heun equation is the starting point for the constructions. The basic technical problem is to choose the bijective relation between the system parameters and the parameters of the deformed confluent Heun equation. The solution of this problem is quite large, and we use the algebraic computing system Maple for this.     

Existence and multiplicity of solutions to even order ordinary differential equations via variational methods

In this paper, we study the existence and multiplicity of solutions for two-point boundary value problems for even order ordinary differential equation. By using the critical point theory and the property of projective operator, we establish some conditions on the nonlinearity which guarantee that the nonlinear Hammerstein integral equation with quasi-positive definite kernel has at least one solution, two nontrivial solutions and finite pairs of solutions, respectively. As an application, we get the corresponding results for two-point boundary value problems for even order ordinary differential equation. Moreover, an example is presented to illustrate our main results.     

含积分边界条件的分数阶微分方程边值问题的正解的存在性

研究了含积分边界条件的分数阶微分方程的边值问题,首先给出格林函数及性质,其次将问题转化为一个等价的积分方程,最后应用Krasnoselkii及Leggett-Williams不动点定理得到了一个及多个正解的存在性,推广了以往的结果．     

Qualitative properties of solutions of volterra equations in Banach spaces

In this paper we consider an abstract Volterra integral equation in an ordered Banach space. We establish some monotonicity properties of solutions and apply these results to their asymptotic behaviour. This is done by using the method of upper and lower solutions.