Fractional Differential Equations

Any physical laws are scale-dependent, the same phenomenon might lead to debating theories if observed using different scales. The two-scale thermodynamics observes the same phenomenon using two different scales, one scale is generally used in the conventional continuum mechanics, and the other scale can reveal the hidden truth beyond the continuum assumption, and fractal calculus has to be adopted to establish governing equations. Here basic properties of fractal calculus are elucidated, and the relationship between the fractal calculus and traditional calculus is revealed using the two-scale transform, fractal variational principles are discussed for 1-D fluid mechanics. Additionally planet distribution in the fractal solar system, dark energy in the fractal space, and a fractal ageing model are also discussed.

http://www.doiserbia.nb.rs/ft.aspx?id=0354-98362000065H

New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle

Solitons, Nonlinear Evolution Equations and Inverse Scattering: Remarks on Riemann-Hilbert problems

5BAn extended tanh-function method is proposed for constructing multiple traveling wave solutions of nonlinear partial differential equations in a unified way. The key idea of this method is to take full advantage of a generalized Riccati equation involving two parameters and use its solutions to replace the tanh function in the tanh-function method. It is quite interesting that the sign of the parameters can be used to exactly judge the numbers and types of these traveling wave solutions.

Extended Tanh-function Method and its Applications to Nonlinear Equations

Variable separation method for a nonlinear time fractional partial differential equation with forcing term

In this paper, a nonisospectral and variable-coefficient KdV equation hierarchy with self-consistent sources is derived from the related linear spectral problem. Exact solutions of the KdV equation hierarchy are obtained through the inverse scattering transformation (IST). It is shown that the IST is an effective mathematical tool for solving the whole hierarchy of nonisospectral nonlinear partial differential equations with self-consistent sources.

Exact solutions of a KdV equation hierarchy with variable coefficients

Purpose – The purpose of this paper is to analytically solve the (2+1)-dimensional nonlinear time fractional biological population model in the Caputo sense. Design/methodology/approach – The paper uses the variable separation method and the properties of Gamma function to construct exact solutions of the time fractional biological population model. Findings – New variable separation solutions are obtained, from which some known solutions are recovered as special cases. Originality/value – Solving fractional biological population model by the variable separation method and the properties of Gamma function is original. It is shown that the method presented in this paper can be also used for some other nonlinear fractional partial differential equations arising in sciences and engineering.

Variable separation method for nonlinear time fractional biological population model

It is well known that the celebrated Kadomtsev-Petviashvili (KP) equation has many important applications. The aim of this article is to use fractional KP equation to not only simulate shallow ocean waves but also construct novel spatial structures. Firstly, the definitions of the conformable fractional partial derivatives and integrals together with a physical interpretation are introduced and then a fractional integrable KP equation consisting of fractional KPI and KPII equations is derived. Secondly, a formula for the fractional - soliton solutions of the derived fractional KP equation is obtained and fractional line one-solitons with bend, wavelet peaks, and peakon are constructed. Thirdly, fractional X-, Y- and 3-in-2-out-type interactions in the fractional line two- and three-soliton solutions of the fractional KPII equation are simulated for shallow ocean waves. Besides, a falling and spreading process of a columnar structure in the fractional line two-soliton solution is also simulated. Finally, a fractional rational solution of the fractional KP equation is obtained including the lump solution as a special case. With the development of time, the nonlinear dynamic evolution of the fractional lump solution of the fractional KPI equation can change from ring and conical structures to lump structure.

/pdf/line-soliton-interactions-for-shallow-ocean-waves-and-novel-edmc1d95o6.pdf

Line Soliton Interactions for Shallow Ocean Waves and Novel Solutions with Peakon, Ring, Conical, Columnar, and Lump Structures Based on Fractional KP Equation

In this paper, the simplest exp-function method which combines the
 exp-function method with a direct algorithm is used to exactly solve the
 Mikhauilov-Novikov-Wang equations. As a result, two explicit and exact
 solutions are obtained. It is shown that the simplest exp-function method
 provides a simpler but more effective mathematical tool for constructing
 exact solutions of non-linear evolution equations in fluids.

/pdf/simplest-exp-function-method-for-exact-solutions-of-3l6rr3l9fu.pdf

Simplest exp-function method for exact solutions of Mikhauilov-Novikov-Wang equations

Kadomtsev-Petviashvili equation is a mathematical model with many important applications in fluids. In this paper, a local fractional Kadomtsev-Petviashvili equation with Lax integrability is derived and solved by extending Hirota?s bilinear method. More specifically, the local fractional Kadomtsev-Petviashvili equation is derived from a local fractional Lax equation. With the help of a suitable transformation, the local fractional Kadomtsev-Petviashvili equation is then bilinearized. Based on the bilinearized form, n-soliton solution with Mittag-Leffler functions is obtained. In order to gain more insights into the fractional n-soliton solution, the velocity of the fractional one-soliton solution is simulated. It is shown that the velocity of the fractional one-soliton changes with the fractional order.

/pdf/bilinearization-and-fractional-soliton-dynamics-of-uvj2wqx50i.pdf

Bo Xu

Papers

Exact solutions of a KdV equation hierarchy with variable coefficients

Variable separation method for nonlinear time fractional biological population model

Line Soliton Interactions for Shallow Ocean Waves and Novel Solutions with Peakon, Ring, Conical, Columnar, and Lump Structures Based on Fractional KP Equation

Simplest exp-function method for exact solutions of Mikhauilov-Novikov-Wang equations

Bilinearization and fractional soliton dynamics of fractional Kadomtsev-Petviashvili equation