New explicit solitons for the general modified fractional Degasperis–Procesi–Camassa–Holm equation with a truncated M-fractional derivative

Important analytical methods such as the methods of exp-function, rational hyperbolic method (RHM) and sec–sech method are applied in this paper to solve fractional nonlinear partial differential equations (FNLPDEs) with a truncated [Formula: see text]-fractional derivative (TMFD), which consist of exponential terms. A general modified fractional Degasperis–Procesi–Camassa–Holm equation (GM-FDP-CHE) is investigated with TMFD. The exp-function method is also applied to derive a variety of traveling wave solutions (TWSs) with distinct physical structures for this nonlinear evolution equation. The RHM is used to obtain single-soliton solutions for this equation. The sec–sech method is used to derive multiple-soliton solutions of the GM-FDP-CHE. These techniques can be implemented to find various differential equations exact solutions arising from problems in engineering. The analytical solution of the [Formula: see text]-fractional heat equation is found. Graphical representations are also given.

New solitary Solution for the Kudryashov-Sinelshchikov (KS) equation by Modern Extension of the Hyperbolic Method

In the present paper, we apply the modern extension of the hyperbolic tanh function method of nonlinear partial differential equations (NLPDEs) of Kudryashov - Sinelshchikov (KS) equation for obtaining exact and solitary traveling wave solutions. Through our solutions, we gain various functions, such as, hyperbolic, trigonometric and rational functions. Additionally, we support our results by comparisons with other methods and painting 3D graphics of the exact solutions. It is shown that our method provides a powerful mathematical tool to find exact solutions for many other nonlinear equations in applied mathematics http://dx.doi.org/10.25130/tjps.25.2020.039

The use of hyperbolic and Asaoka’s methods to estimate the moisture diffusivity and the hydraulic conductivity of unsaturated soils from tests to determine the SWRC

The equation governing the unsaturated transient flow in a soil sample when subjected to suction (ψ) at its base and that of the classical 1-D consolidation are exactly alike (equation of diffusion). The former can be arrived at by Richards’ equation, being in this case the moisture diffusivity (D), instead of the coefficient of consolidation (cv), the governing parameter. D need not be constant, but rather a function of the volumetric water content, D=D(θ), and is defined as the ratio of the hydraulic conductivity, k(θ), over the specific water capacity, C(θ)=dθ/dψ, i.e., the slope of the SWRC. The hyperbolic method has been used for several geotechnical purposes and, most importantly, as an alternative to Asaoka’s method for predicting the final settlement and cv of soft soils undergoing consolidation, improved by preloading. This paper shows that both methods prove to be very useful as a means of obtaining D(θ) and k(θ) at a certain range of θ, provided that a reduced number of water contents at known elapsed times are determined over the medium stage of this transient flow. It is addressed in the paper both by theoretical grounds and on the light of experimental data of 4 soils.