For the newly implemented 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equation family, the present study explores the exact singular, solitary, and periodic singular wave solutions via the (G′/G2)-expansion process. In the sense of conformable derivatives, the equations considered are translated into ordinary differential equations. In spite with many trigonometric, complex hyperbolic, and rational functions, some fresh exact singular, solitary, and periodic wave solutions to the deliberated equations in fractional systems are attained by the implementation of the (G′/G2)-expansion technique through the computational software Mathematica. The unique solutions derived by the process defined are articulated with the arrangement of the functions tanh, sech; tan, sec; coth, cosech, and cot, cosec. With three-dimensional (3D), two dimensional (2D) and contour graphics, some of the latest solutions created have been envisaged, selecting appropriate arbitrary constraints to illustrate their physical representation. The outcomes were obtained to determine the power of the completed technique to calculate the exact solutions of the equations of the WBBM that can be used to apply the nonlinear water model in the ocean and coastal engineering. All the solutions given have been certified by replacing their corresponding equations with the computational software Mathematica.

In this current study, we described a modified extended tanh-function (mETF) method to find the new and efficient exact travelling and solitary wave solutions to the modified Liouville equation and modified regularized long wave (mRLW) equation in water wave mechanics. Travelling wave transformation decreases the leading equation to traditional ordinary differential equations (ODEs). The standardized balance technique provides the instruction of the portended polynomial related result stimulated from the mETF method. The substitution of this result follows the preceding step. Balancing the coefficients of the like powers of the portended solution leads to a system of algebraic equations (SAE). The solution of that SAE for coefficients provides the essential connection between the coefficients and the parameters to build the exact solution. Here the acquired solutions are hyperbolic, rational, and trigonometric function solutions. Our mentioned method is straightforward, succinct, efficient, and powerful and can be emphasized to establish the new exact solutions of different types of nonlinear conformable fractional equations in engineering and further nonlinear treatments.

In this article, we consider the exact solutions to the simplified modified Camassa-Holm (SMCH) equation which has many potential applications in mathematical physics and engineering sciences. We examine the exact travelling wave solutions by means of the modified simple equation (MSE) method by making use of travelling transformation. The attained solutions are in the form of trigonometric and hyperbolic functions. We demonstrate that the method is more general, straightforward and powerful and can be used to examine more general travelling wave solutions of various kinds of fractional nonlinear differential equations arising in mathematical physics and better than other method. Finally, we show the graphical representation and discuss the physical significance of the obtained solutions for its definite values of the involved parameters through depicting 3D and 2D figures in order to know the physical phenomena.

This article aims to construct the presence and diversity principles of minimum one or two positive solutions for a Caputo-type fractional-order nonlinear differential equation (CFONLDE for short) with an advanced argument under three-point boundary value conditions (BVCs for short). Guo-Krasnoselskii's fixed point theorem and Fixed-point index theory in cone spaces are used to analyze this article. First, the Green's function of the corresponding boundary value problem for a linear fractional differential equation with an advanced argument has been established. Next, several essential properties of that Green's function have been proved. Finally, in cone spaces, some novel presence and diversity principles of minimum of one or two positive solutions for a CFONLDE with an advanced argument are obtained. To support the analytic proof, some particular examples are included.

In this paper, we investigate the existence criteria of at least one positive solution to the three-point boundary value problems with coupled system of Riemann-Liouville type nonlinear fractional order differential equations. The analysis of this study is based on the well-known Schauder's fixed point theorem. Some new existence and multiplicity results for coupled system of Riemann-Liouville type nonlinear fractional order differential equation with three-point boundary value conditions are obtained.