Fractional Differential Equations

Applied Numerical Analysis. By C.F. Gerald. Pp. xii, 340. £3·60. 1970. (Addison-Wesley.)

The main purpos e of this chapter is to present a direct and systematic way of finding exact solutions and Backlund transformations of a certain class of nonlinear evolution equations. The nonlinear evolution equations are transformed, by changing the dependent variable(s), into bilinear differential equations of the following special form 
 
$$ F\left( {\frac{\partial }{{\partial t}} - \frac{\partial }{{\partial {t^1}}},\frac{\partial }{{\partial x}} - \frac{\partial }{{\partial {x^1}}}} \right)f(t,x)f({t^1},{x^1}){|_{t = {t^1},x = {x^1}}} = 0 $$ 
 
, which we solve exactly using a kind of perturbational approach.

Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告)

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

With symbolic computation, two classes of lump solutions to the dimensionally reduced equations in (2+1)-dimensions are derived, respectively, by searching for positive quadratic function solutions to the associated bilinear equations. To guarantee analyticity and rational localization of the lumps, two sets of sufficient and necessary conditions are presented on the parameters involved in the solutions. Localized characteristics and energy distribution of the lump solutions are also analyzed and illustrated.

Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation

In this paper, the first integral method is used to construct exact solutions of the time-fractional Wu---Zhang system. Fractional derivatives are described by conformable fractional derivative. This method is based on the ring theory of commutative algebra. The results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear conformable time-fractional partial differential equations.

The first integral method for Wu---Zhang system with conformable time-fractional derivative

In this paper, fractional derivatives in the sense of modified Riemann-Liouville derivative and first integral method are applied for constructing exact solutions of nonlinear fractional generalized reaction duffing model and nonlinear fractional diffusion reaction equation with quadratic and cubic nonlinearity. Our approach provides first integrals in polynomial form with a high accuracy for two-dimensional plane autonomous systems. Exact soliton solutions are constructed through established first integrals.

Application of first integral method to fractional partial differential equations

This paper obtains soliton solutions to optical couplers by two methods. These are sine–cosine function method and Bernoulli’s equation approach. There are four laws that are touched upon in this paper. These are Kerr law, power law, parabolic law and dual-power law. The first integration scheme is applicable to Kerr and power laws only where bright soliton solutions are retrievable. The second tool is applicable to parabolic and dual-power laws only that leads to dark and singular solitons for these two nonlinear media.

Optical solitons in nonlinear directional couplers by sine–cosine function method and Bernoulli’s equation approach

Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations

This paper applied the trial solution technique to chiral nonlinear Schrodinger’s equation in (1
 $$+$$
 2)-dimensions. This led to solitons and other solutions to the model. Besides soliton and singular soliton solutions, this integration scheme also gave way to singular periodic solutions.

Mostafa Eslami

Papers

The first integral method for Wu---Zhang system with conformable time-fractional derivative

Application of first integral method to fractional partial differential equations

Optical solitons in nonlinear directional couplers by sine–cosine function method and Bernoulli’s equation approach

Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations

Trial solution technique to chiral nonlinear Schrodinger’s equation in (1+2)-dimensions