Abstract In this paper, the differential transform method is used to find approximate analytical and numerical solutions of singularperturbationproblems. The principle of the method is briefly introduced and then applied for solving two mathematical models of stiff initial value singularperturbationproblems. 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 [–]
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 problemssingular in the dependent variable. In: Handbook of Diﬀerential Equations: Ordinary Diﬀerential Equations, vol. 1, pp. 1-68 (2000)
Abel-type integralequations 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 integralequations by using various analysis methods (see [–] and references therein).
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:
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
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)
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 nonlinearequations 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 nonlinearequations 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].
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 singularintegral 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 nonlinearsingularintegralequations consist with the latest high technology for the solution of modern problems of solid mechanics, ﬂuid mechanics, and aerodynamics.
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
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.
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.
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
A new integral transform is derived from the classical Fourier integral. A new integral transform  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.
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 .
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]
The successful development of the theory of singularintegralequations (SIE) naturally stimulated the study of singularintegralequations with shift (SIES). (see [9,11,13,14], [15-18] and others). Existence results and approximate solutions have been studied for certain classes of nonlinearsingularintegralequations (NSIE) and nonlinearsingularintegralequations 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 ). The theory of SIES is an important part of integralequations because of its recent applications in many fields of physics and engineering, (see [8,15,17]).