# Top PDF Solutions of nonlinear integral equations and their application to singular perturbation problems ### Solutions of nonlinear integral equations and their application to singular perturbation problems

Wasow  proves the existence of a solution and develops a single uniform asymptotic expansion for that solution in the case of a second order differential equation of the form 1.3 wit[r] ### An Application of Differential Transform Method in Singular Perturbation Problems

The aim of our study is to use the Differential Transform Method (DTM) as an alternative to existing methods for solving this class of SPPs. The DTM was first introduced by Zhou  and its main application concern with both linear and nonlinear initial value problems in electrical circuit analysis. This method constructs a semi-analytical numerical technique that uses Taylor series for the solution of differential equations in the form of a polynomial. However, it is different from traditional higher order Taylor series method, which requires symbolic computation of the necessary derivatives of the data function. The main advantage of this method is that it can be applied directly to nonlinear ODEs without requiring linearization, discretization or perturbation. Another important advantage is that this method is capable of greatly reducing the size of computational work while still accurately providing the series solution with fast convergence rate. Different applications of DTM can be found in [3-14,17]. In this paper, we will apply DTM to find approximate analytical and numerical solutions of a class of singular perturbation problems. The principle of the method is briefly introduced and then applied to a particular case of stiff initial value singular perturbation problems. The results are then compared with the exact solutions to demonstrate the reliability and efficiency of the method in solving the considered problems. ### Positive solutions for singular boundary value problems involving integral conditions

Integral boundary conditions and multi-point boundary conditions for diﬀerential equations come from many areas of applied mathematics and physics [–]. Recently, sin- gular boundary value problems have been extensively considered in a lot of literature [, , , ], since they model many physical phenomena including gas diﬀusion through porous media, nonlinear diﬀusion generated by nonlinear sources, chemically reacting systems as well as concentration in chemical or biological problems. In all these problems, positive solutions are very meaningful. ### Analysis of Abel-type nonlinear integral equations with weakly singular kernels

Abel-type integral equations are associated with a wide range of physical problems such as heat transfer [], nonlinear diﬀusion [], propagation of nonlinear waves [], and they can also be applied in the theory of neutron transport and in traﬃc theory. In the past  years, many researchers investigated the existence and uniqueness of nontrivial solutions for a large number of Abel-type integral equations by using various analysis methods (see [–] and references therein). ### Solvability for system of nonlinear singular differential equations with integral boundary conditions

In recent years, to the best of our knowledge, although there are many papers concern- ing the existence of positive solutions for nth order boundary value problems with dif- ferent kinds of boundary conditions for system (see [–] and the references therein), results for the system (.) are rarely seen. Moreover, the methods mainly depend on the Krasonsel’skii ﬁxed point theorem, ﬁxed point index theory, the upper and lower solution technique, some new ﬁxed point theorem for cones, etc. For example, in [], by applying the Krasonsel’skii ﬁxed point theorem, Henderson and Ntouyas studied the existence of at least one positive solution for the following system: ### Nonlinear unsteady flow problems by multidimensional singular integral representation analysis

1. Introduction. Over the last years, an increasing interest has been con- centrated on the study of nonlinear multidimensional singular integral equa- tions because of their application to the solution of modern and complicated problems of solid and ﬂuid mechanics theory. Such problems are solved by computational methods as closed-form solutions are not possible to be de- termined. The algorithms which are used for the numerical evaluation of the nonlinear singular integral equations consist with the latest high technology for the solution of modern problems of solid mechanics, ﬂuid mechanics, and aerodynamics. ### Positive solutions for singular nonlinear fractional differential equation with integral boundary conditions

integral with a signed measure, in which A : [, ] → R is a function of bounded variation. Fractional diﬀerential equations have attracted more and more attention from the re- search communities due to their numerous applications in many ﬁelds of science and engineering including ﬂuid ﬂow, rheology, diﬀusive transport akin to diﬀusion, electri- cal networks, probability, etc. For details, see [–] and the references therein. On the other hand, boundary value problems with integral boundary conditions for ordinary dif- ferential equations arise in many ﬁelds of applied mathematics and physics such as heat conduction, chemical engineering, underground water ﬂow, thermoelasticity, and plasma physics. The existence and multiplicity of positive solutions for such problems have be- come an important area of investigation in recent years. ### Existence of multiple positive solutions for singular boundary value problems of nonlinear fractional differential equations

t ∈ J := [0, 1], u(0) = u (1) = 0, where 1 < q ≤ 2 is a real number, and D q 0+ is the standard Riemann-Liouville diﬀerentiation. As an application of the Green’s function, we give some multiple positive solutions for singular boundary value problems, and we also give the uniqueness of solution for a singular problem by means of the ### 1.Introduction In recent years, singular integral equations arise in many problems of

In Table 2, we use Maple programm to compute the exact and approximate solutions and the errors Ε N of Eq .(5.3) , (5.4) with logarithmic kernel numerically by using the product Nyström method , for linear (k =1) and nonlinear (k = 3) cases, , for different values of υ = 0.389 and 0.22, that corresponding to Polyurethane and Nickel materials at T= 0.1, 0.5 ,0.9, and N = 20, 40 ,. Also, the error Ε N is plotted as shown in Figs. (1-7) -(1-12) ### Multiple positive solutions to singular positone and semipositone m-point boundary value problems of nonlinear fractional differential equations

nonlinearity f may be singular at u = 0. As an application of Green’s function, we give some multiple positive solutions for singular positone and semipositone boundary value problems by means of the Leray–Schauder nonlinear alternative and a ﬁxed point theorem on cones. ### Bright and Singular Soliton Solutions of Tzitzéica Type Equations Appear in Nonlinear Optics

Parallel developments in both computer technologies and symbolic softwares have greatly contributed to solve lots of problems defined in various fields covering applied mathematics, physics and many engineering fields. A diverse class of effective methods have successfully been introduced to study this class of equations, for example [3, 10–15]. On the other hand some of the commonly used approaches, for solving nonlinear evolution equations, are: The ansatz [16–18], modified simple equation , the first integral [20,21], ( G G 0 )-expansion , sine-Gordon expansion [23, 24]. Furthermore, some other excellent works like Kudryashov methods , a modified form of Kudryashov and functional variable methods [26–28] have been done by different researchers. In [29–32], the auxiliary equation, the improved tan( φ(η) 2 )-expansion methods and the exp function approach have been explored for discrete and fractional order PDEs as well. Ali and Hassan , Hosseini et al.  and Zayed and Al-Nowehy  all have utilized the exp a function method to explore the exact solutions ### Iterative positive solutions for singular nonlinear fractional differential equation with integral boundary conditions

In this article, we study the existence of iterative positive solutions for a class of singular nonlinear fractional diﬀerential equations with Riemann-Stieltjes integral boundary conditions, where the nonlinear term may be singular both for time and space variables. By using the properties of the Green function and the ﬁxed point theorem of mixed monotone operators in cones we obtain some results on the existence and uniqueness of positive solutions. We also construct successively some sequences for approximating the unique solution. Our results include the multipoint boundary problems and integral boundary problems as special cases, and we also extend and improve many known results including singular and non-singular cases. ### Uniqueness of iterative positive solutions for the singular fractional differential equations with integral boundary conditions

In the past couple of decades, boundary value problems for nonlinear fractional diﬀer- ential equations arise from the studies of complex problems in many disciplinary areas such as aerodynamics, ﬂuid ﬂows, electrodynamics of complex medium, electrical net- works, rheology, polymer rheology, economics, biology chemical physics, control theory, signal and image processing, blood ﬂow phenomena, and so on. Fractional-order models have been shown to be more accurate and realistic than integer-order models, and with this advantage in the application of these models, it is important to theoretically estab- lish the conditions for the existence of positive solutions because theoretical results can help people to get an in-depth understanding for the dynamic behavior in the practical process, so the study of abstract fractional models is important and relevant nowadays. In recent years, many authors investigated the existence of positive solutions for fractional equation boundary value problems (see [–] and the references therein), and a great deal of results have been developed for diﬀerential and integral boundary value problems. The authors in [] studied the following system of singular fractional diﬀerential equa- ### Exact solutions of nonlinear interval Volterra integral equations

There are many applications in real problems that modeled by Volterra integral equa- tions, for example, in ﬂuid mechanics, bio-mechanics [16, 24, 26]. There are many methods both numerical and analytical approaches to solve nonlinear equations like as ﬁnite diﬀer- ence method, ﬁnite element method, homotopy analysis method, homotopy perturbation method and variational iteration method and its modiﬁcation [1, 2, 3, 6, 5, 8, 9, 10, 12, 25]. One of the eﬃcient tool for solving nonlinear equations is the decomposition method which was stated in [4, 15]. In this direction, Khan et al. [14, 18] have improved this method. This improvement has the following main advantageous that we can select initial guess appropriately without having noise terms. Also, some eﬀorts have been done in [17, 19, 20]. ### Positive solutions for singular coupled integral boundary value problems of nonlinear higher order fractional q difference equations

Research on q-diﬀerence calculus or quantum calculus dates back to the beginning of the th century, when Jackson [, ] introduced the ﬁrst deﬁnition of the q-diﬀerence. Then Al-Salam [] and Agarwal [] proposed the fractional q-diﬀerence calculus. Later, the theory of fractional q-diﬀerence calculus itself and nonlinear fractional q-diﬀerence equation boundary value problems have been extensively studied by many authors. For some recent developments on fractional q-diﬀerence calculus and boundary value prob- lems of fractional q-diﬀerence equations, see [–] and the references therein. For ex- ample, by applying the generalized Banach contraction principle, the monotone iterative method, and Krasnoselskii’s ﬁxed point theorem Zhao et al. [] showed some existence results of positive solutions to nonlocal q-integral boundary value problem of nonlinear fractional q-derivatives equation. Under diﬀerent conditions, Graef and Kong [, ] in- vestigated the existence of positive solutions for boundary value problems with fractional q-derivatives in terms of diﬀerent ranges of λ, respectively. By applying some standard ﬁxed point theorems, Agarwal et al. [] and Ahmad et al. [] showed some existence re- sults for sequential q-fractional integrodiﬀerential equations with q-antiperiodic bound- ary conditions and nonlocal four-point boundary conditions, respectively. ### Existence and approximate solutions of nonlinear integral equations

t,s, x(s) ds, −∞ ≤ a ≤ t ≤ b ≤ + ∞ . (1.2) We should mention that an extensive work has been done in the study of the solutions of various types of (1.2), see, for example, [1, 2, 5, 7, 11, 13, 15, 16, 17, 19]. Usually the existence of a solution of (1.2) starts with some conditions on the function g(t,s,x) as well as the integration bounds a, b and the function f ( · ). Based on these conditions, a Banach space is chosen in such a way that the existence problem is converted to a fixed point problem of an operator over this Banach space. ### Review of singular potential integrals for method of moments solutions of surface integral equations

Exact solutions of electromagnetic antenna and scattering problems often rely on integral equations being solved by the method of moments (MoM) with Galerkin’s method, us- ing triangular Rao-Wilton-Glisson (RWG) vector basis func- tions (Rao et al., 1982). In mixed potential formulations for metallic and dielectric scatterers, the kernels of surface inte- grals include the Green’s function of free space, as well as the gradient of Green’s function. Thus, singularities of order 1/R and ∇ 1/R must be considered, where R =| r − r 0 | is the distance between observation and source points. Because of these terms, the surface integrals become singular if a test- ing point r is near the source element. In order to calculate the singular surface integrals, special methods must be used, because numerical integration routines lead to inaccurate so- lutions. There are many methods that can be used to evaluate singular potential integrals, such as the Duffy’s transforma- Correspondence to: A. Tzoulis ### Existence of solutions to a class of quasilinear elliptic problems with nonlinear singular terms

[] not only improved the results in [] but also obtained that the solution was not in W  ,p () if α > p– p– . However, we need to point out that all the papers mentioned discussed the existence of solutions by means of upper-lower solution techniques. In this paper, we apply the method of regularization and Schauder’s ﬁxed point theorem as well as a nec- essary compactness argument to overcome some diﬃculties arising from the nonlinearity of the diﬀerential operator, the singularity of nonlinear terms and the summability of the weighted function f (x) and then prove the existence of positive solutions in W  ,p () for  