boundary layer are given for x and x 1 • It is characteristic of these asymptotic expansions to consist of parts that may be computed by using only ordinary pert[r]

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Abstract In this paper, the differential transform method is used to find approximate analytical and numerical **solutions** of **singular** **perturbation** **problems**. The principle of the method is briefly introduced and then applied for solving two mathematical models 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**.

Therefore, there have been some papers dealing with the existence and multiplicity of so- lutions or positive **solutions** for boundary value **problems** involving **nonlinear** fractional diﬀerential **equations**; see [–] and references cited therein.
At the same time, we notice that boundary value **problems** for a coupled system of non- linear fractional diﬀerential **equations** have been addressed by several researchers. For instance, for some results for the existence of **solutions** or positive **solutions** for a cou- pled system of **nonlinear** fractional diﬀerential **equations**, we refer the readers to [–]

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19. Agarwal, RP, O’Regan, D: Existence theory for single and multiple **solutions** to **singular** positone boundary value **problems**. J. Diﬀer. Equ. 175, 393-414 (2001)
20. Agarwal, RP, O’Regan, D: A survey of recent results for initial and boundary value **problems** **singular** in the dependent variable. In: Handbook of Diﬀerential **Equations**: Ordinary Diﬀerential **Equations**, vol. 1, pp. 1-68 (2000)

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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).

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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:

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In this paper, we consider the properties of the Green’s function for the **nonlinear** fractional diﬀerential equation boundary value problem D q 0+ u(t) = f (t,u(t)),
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

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Adv. Diﬀer. Equ. 2012, 66 (2012)
11. Hao, X., Liu, L., Wu, Y.: Positive **solutions** for **nonlinear** fractional semipositone diﬀerential equation with nonlocal boundary conditions. J. **Nonlinear** Sci. Appl. 9, 3992–4002 (2016)
12. Jebari, R.: Solvability and positive **solutions** of a system of higher order fractional boundary value problem with **integral** conditions. Fract. Diﬀer. Calc. 6, 179–199 (2016)

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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].

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2000 Mathematics Subject Classiﬁcation: 65L10, 65R20.
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.

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µθ − λ ∫ − − θ = , ( 0 < < υ 1 ) . (5.2 T he N um erical R esults T he N um erical R esults T he N um erical R esults: T he N um erical R esults
Maple programm is used to compute the exact and approximate **solutions** and errors Ε N of Eqs.(5.1), (5.2) with Carleman kernel by using the product Nyström methods 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

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h(s)u(s) dA(s) denotes the Riemann-Stieltjes **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.

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7. Composite Feedback Control of **Nonlinear** Systems
In the preceding three sections approximations of both the optimal feedback control and the optimal trajectory consisted of slow and fast parts.
They were obtained from singularly perturbed Riccati **equations** or two-point boundary value **problems**. These optimality conditions also consisted of slow and fast parts. A further step toward a final decomposition of the two-time- scale design has been made which decomposes the optimal control problem itself into a slow subproblem and a fast subproblem. Separate **solutions** of these

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He’s Polynomials
1. Introduction
A new **integral** transform is derived from the classical Fourier **integral**. A new **integral** transform [3] was intro- duced by Artion Kashuri and Associate Professor Akli Fundo to facilitate the process of solving ordinary and partial differential **equations** in the time domain. Some **integral** transform method such as Laplace, Fourier, Su- mudu and Elzaki transforms methods, are used to solve general **nonlinear** non-homogenous partial differential equation with initial conditions and use fullness of these **integral** transform lies in their ability to transform dif- ferential **equations** into algebraic **equations** which allows simple and systematic solution procedures. Non-linear phenomena, that appear in many areas of scientific fields such as solid state physics, plasma physics, fluid me- chanics, population models and chemical kinetics, can be modeled by **nonlinear** differential **equations**. The impor- tance of obtaining the exact or approximate **solutions** of **nonlinear** partial differential **equations** in physics and mathematics is still a significant problem that needs new methods to discover exact or approximate **solutions**. Also a new **integral** transform and some of its fundamental properties are used to solve general **nonlinear** non-homo- genous partial differential equation with initial conditions.

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Soliton and periodic **solutions** were derived by the hyperbolic tangent method together with the Painleve property to both the TDBM (1) and the TT (2) **equations** [3].
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 [19], the first **integral** [20,21], ( G G 0 )-expansion [22], sine-Gordon expansion [23, 24]. Furthermore, some other excellent works like Kudryashov methods [25], a modified form of Kudryashov and functional variable methods [26–28]

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Uniformly Valid Asymptotic Solution to a Volterra Equation on an infinite Interval,. Integral Equations of Volterra Type, J.[r]

The successful development of the theory of **singular** **integral** **equations** (SIE) naturally stimulated the study of **singular** **integral** **equations** with shift (SIES). (see [9,11,13,14], [15-18] and others). Existence results and approximate **solutions** have been studied for certain classes of **nonlinear** **singular** **integral** **equations** (NSIE) and **nonlinear** **singular** **integral** **equations** with shift (NSIES) by many authors among them we mention [1-6, 12, 20]. The classical and more recent results on the solvability of NSIE should be generalized to corresponding **equations** with shift,(see [22]). The theory of SIES is an important part of **integral** **equations** because of its recent applications in many fields of physics and engineering, (see [8,15,17]).

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Ruy, On a nonexistence of positive solution of Laplace equation in upper half-space with Cauchy data, Demonstratio Mathematica 28 (1995), no. Binh, On a nonexistence of positive solution[r]

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There is a large literature on nonlinear singular integral equations with Hiibert and Cauchy kernel and on related nonlinear Riemann-Hilbert problems for analytic functions, cf.. the mon[r]