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1 Introduction Mathematical modelling of physical phenomena in all branches of science and engineering frequently results in boundary value problems governed by nonlinear differential equations. These problems generally do not have closed-form, exact analytical solutions. However, because of their many important applications, significant efforts have been and continue to be made to accurately approximate solutions to these problems. Early methods developed to approximate solutions to nonlinear boundary value problems (BVPs) were strictly numerical. Whereas

Georgian Mathematical Journal Volume 13 (2006), Number 2, 215–228 EXISTENCE THEORY FOR PERTURBED NONLINEAR BOUNDARY VALUE PROBLEMS WITH INTEGRAL BOUNDARY CONDITIONS ABDELKADER BELARBI, MOUFFAK BENCHOHRA, AND BAPURAO C. DHAGE Abstract. In this paper, the existence of solutions and extremal solutions for a second order perturbed nonlinear boundary value problem with integral boundary conditions is proved under the mixed generalized Lipschitz and Carathéodory conditions. 2000 Mathematics Subject Classification: 34A60, 34B15. Key words and phrases: Nonlinear

[1] BERNFELD, S. R.— LAKSHMIKANTHAM, V.: An Introduction to Nonlinear Boundary Value Problems, Academic Press Inc., New York-London, 1974. [2] BAILEY, P. B.— SHAMPINE, L. F.— WALTMAN, P. E.: Nonlinear Two Point Boundary Value Problems, Academic Press Inc., New York-London, 1968. [3] DINCA, G.— SANCHEZ, L.: Multiple solutions of boundary value problems: an elementary approach via the shooting method, NoDEA Nonlinear Differential Equations Appl. 1 (1994), 163–178. http://dx.doi.org/10.1007/BF01193950 [4] FUČÍK, S.— KUFNER, A.: Nonlinear Differential Equations

1 Introduction Many problems in science and engineering can be reduced to the problem of solving nonlinear boundary value problems. For example, it is well known that fourth-order two-point nonlinear boundary value problems are essential in describing a large class of elastic deflections. Multiplicity results for a fourth-order nonlinear boundary value problem modeling elastic beams are presented in Ref. [ 1 ]. The boundary value problem of torsion of a solid cylinder is analyzed for a class of hyperelastic materials that exhibit the power law type dependence of

are obtained. In section 3 , the proposed method is applied for solving some nonlinear boundary value problems in the semiinfinite domain. Finally, a brief conclusion is given in the last section. 2 Generalized Fractional order of the Chebyshev Functions In this section, first, the generalized fractional order of the Chebyshev functions (GFCFs) of the first kind have been defined and then some properties and convergence of them for our method have been provided. 2.1 The GFCFs definition The efficient methods have been used by many researchers to solve the

References [1] Agarwal R.P., O’Regan D., Wong P.J.Y., Positive solutions of differential, difference and integral equations, Kluwer, Dordrecht, 1999. [2] Wazwaz A.M., A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems, Comput. Math. Appl., 2001, 41, 1237–1244. [3] Duan J.S., Rach R., A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations, Appl. Math. Comput., 2011, 218, 4090–4118. [4] Adomian G., Rach R., Meyers R., Numerical

Chapter 2 Solvability of nonlinear boundary value problems 2.1 Formulation of the problem Let Ω be a domain in R N with boundary 3Ω. We will consider the following (partial) differential operator of order 2k in the divergence form: (Au)(x)= (2.1) l«l<* (see Introduction, formula (0.22)) where aa = aa(x\ ξ) are functions defined on Ω Χ Mm, aa G CAR. Further, let w = {wa (x); |o;| < k} be a family of weight functions on Ω satisfying the conditions wa G Ll]0C(Q), w~7=1 G L /^ ίΩ) ,\a\<k, (2.2) with some parameter ρ > 1, and let us consider the weighted

On the solution of some nonlinear boundary value problem Gabriella Bogndr Abstract. The subject of this paper is to give classical solutions for the Dirichlet and Neumann problem to the quasilinear equation of the divergence form d (du\"p d fdu\p - n . „ „2 * where ρ > 0 real, the function υ ρ is defined by f vp if ν > 0, VP = < I —|u|p if ν < 0, and Ω is bounded by a rectangle. We give the eigenfunctions and the corresponding eigenvalues moreover an asymptotic formula for the eigenvalues in both cases when the Dirichlet and when the Neumann condition

3th Int. Coll. on Numerical Analysis, pp. 15-24 D. Bainov and V. Covachev (Eds) © VSP 1995 Numerical Solution of Singular Nonlinear Boundary Value Problems John V. Baxley Department of Mathematics and Computer Science Wake Forest University Winston-Salem, NC 27106 USA Abstract: We discuss numerical procedures for boundary value prob- lems for nonlinear ordinary equations with singularities at the endpoints, of the type y " + ^ = 0, 0 < f < l , y y( 0) = 0, y ( l ) = 0, where (f){t) is positive and continuous on 0 < t < 1 and A > 0. Our methods are

Abstract

In this article, we discuss the nonlinear boundary value problems involving both left Riemann-Liouville and right Caputo-type fractional derivatives. By using some new techniques and properties of the Mittag-Leffler functions, we introduce a formula of the solutions for the aforementioned problems, which can be regarded as a novelty item. Moreover, we obtain the existence result of solutions for the aforementioned problems and present the Ulam-Hyers stability of the fractional differential equation involving two different fractional derivatives. An example is given to illustrate our theoretical result.