Fractional Differential Equations

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Nonlinear Functional Analysis And Its Applications

This review is an elementary introduction to the concepts of the recently developed asymptotic methods and new developments. Particular attention is paid throughout the paper to giving an intuitive grasp for Lagrange multiplier, calculus of variations, optimization, variational iteration method, parameter-expansion method, exp-function method, homotopy perturbation method, and ancient Chinese mathematics as well. Subsequently, nanomechanics in textile engineering and E-innit y theory in high energy physics, Kleiber’s 3/4 law in biology, possible mechanism in spider-spinning process and fractal approach to carbon nanotube are briey introduced. Bubble-electrospinning for mass production of nanob ers is illustrated. There are in total more than 280 references.

An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering

Differential equations with non-standard growth have been a very active field of investigation in recent years. In this survey we present an overview of the field, as well as several of the most important results. We consider both existence and regularity questions. Finally, we provide a comprehensive list of papers published to date.

Overview of differential equations with non-standard growth

This paper is an elementary introduction to some new asymptotic methods for the search for the solitary solutions of nonlinear differential equations, nonlinear differential-difference equations, and nonlinear fractional differential equations. Particular attention is paid throughout the paper to giving an intuitive grasp for the variational approach, the Hamiltonian approach, the variational iteration method, the homotopy perturbation method, the parameter-expansion method, the Yang-Laplace transform, the Yang-Fourier transform, and ancient Chinese mathematics. Hamilton principle and variational principles are also emphasized. The reviewed asymptotic methods are easy to be followed for various applications. Some ideas on this paper are first appeared.

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Asymptotic Methods for Solitary Solutions and Compactons

Application of variational iteration method and homotopy–perturbation method for nonlinear heat diffusion and heat transfer equations

We investigate the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem -Au(x) = Ag(x)u(x) onD; , (x) +au(x) 0on D a (x) + au(x) = 0 on AD, where D is a bounded region in RN with smooth boundary, g: D -R is a smooth function which changes sign on D, and a E R. Such problems have been studied in recent years because of associated nonlinear problems arising in the study of population genetics (see [3]). The study of the linear ordinary differential equation case, however, goes back to Picone and Bocher (see [2]). Attention has been confined mainly to the cases of Dirichlet (a = oc) and Neumann boundary conditions. In the case of Dirichlet boundary conditions it is well known (see [4]) that there exists a double sequence of eigenvalues for (1)> * **\> 0); in the case where fD g(x) = 0 there are no positive and no negative principal eigenvalues. We shall investigate how the principal eigenvalues of (1), depend on a, obtaining new results for the case where a 0, probably because it is more natural that the flux across the boundary should be outwards if there is a positive concentration at the Received by the editors April 30, 1997. 1991 Mathematics Subject Classification. Primary 35J15, 35J25.

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On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions

On a Diffusive Logistic Equation

Application of homotopy-perturbation method to the second kind of nonlinear integral equations

In this paper, we prove the existence of at least three weak solutions for the quasilinear elliptic systems { Δ p 1 u 1 + λ F u 1 ( x , u 1 , u 2 , … , u n ) = 0 in  Ω , Δ p 2 u 2 + λ F u 2 ( x , u 1 , u 2 , … , u n ) = 0 in  Ω , … Δ p n u n + λ F u n ( x , u 1 , u 2 , … , u n ) = 0 in  Ω , u i = 0  for  1 ≤ i ≤ n on  ∂ Ω . Our main tool is a recent three critical points theorem of Ricceri [B. Ricceri, On a three critical points theorem, Arch. Math. (Basel) 75 (2000) 220–226].

Ghasem A. Afrouzi

Papers

Application of variational iteration method and homotopy–perturbation method for nonlinear heat diffusion and heat transfer equations

On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions

On a Diffusive Logistic Equation

Application of homotopy-perturbation method to the second kind of nonlinear integral equations

Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the (p1,…,pn) -Laplacian