Exact solutions to the space-time fraction Whitham–Broer–Kaup equation

The objective work of this paper is to transform the nonlinear space-time fraction Whitham–Broer–Kaup equation into ordinary differential equation by using the conformal fractional derivative, and find the exact solutions through the complete polynomial discriminant system. At the same time, we build the appropriate solution for the identified parameters to show the existence of the solution. In addition, we provide the 3D and 2D graphics to show that the solutions are real and effective.

Lower bounds for discriminants of polynomials

Let [Formula: see text] be a monic irreducible polynomial of degree [Formula: see text] having exactly [Formula: see text] real roots and [Formula: see text] complex roots with integer coefficients and [Formula: see text] be its polynomial discriminant. Suppose that [Formula: see text] is square-free. In this paper, we will give a lower bound for polynomial discriminants [Formula: see text] for given [Formula: see text].

Double Points and Unstable Configurations of Stephenson-I, -II, and -III Mechanisms

Double points are conjugate configurations in mechanisms, with multiple closed-loops, when both input and output links are fixed in the same position and some passive joints can be located in two different configurations. Unstable configurations, on other hand, are positions where a mechanism loses controllability and gains at least one unwanted DOF instantaneously. The analytical condition for the occurrence of double point is the same as the occurrence of an unstable configuration, i.e., two roots of input-output displacement polynomial are equal. This paper addresses the determination of double points and unstable configurations of six-link Stephenson-I, -II and -III mechanisms. The double points are determined by using the loop-closure equations for two branches and successively eliminating intermediate joint variables using closed-form techniques. Further, extraneous roots from algebraic manipulations are eliminated using a new technique of two-branch equation substitution. The unstable configuration polynomial is derived by (i) successively eliminating intermediate variables from loop-closure equations to obtain the input-output displacement polynomial, (ii) equating the polynomial discriminant to zero to obtain a polynomial which contains both unstable configurations and double points, and (iii) eliminating extraneous roots from algebraic manipulations and double points from this composite polynomial to determine the unstable configurations. The computational procedure is illustrated through numerical examples.

Double Points and Unstable Configurations of Watt-I and Watt-II Mechanisms

Double points are conjugate configurations in mechanisms, with multiple closed-loops, when both input and output links are fixed in the same position and some passive joints can be located in two different configurations. Unstable configurations, on other hand, are positions where a mechanism loses controllability and gains at least one unwanted DOF instantaneously. The analytical condition for the occurrence of double point is the same as the occurrence of an unstable configuration, i.e., two roots of input-output displacement polynomial are equal. This paper addresses the determination of double points and unstable configurations of six-link Watt-I and Watt-II mechanisms. The double points are determined by using the loop-closure equations for two branches and successively eliminating intermediate joint variables using closed-form techniques. Further, extraneous roots from algebraic manipulations are eliminated using a new technique of two-branch equation substitution. The unstable configuration polynomial is derived by (i) successively eliminating intermediate variables from loop-closure equations to obtain the input-output displacement polynomial, (ii) equating the polynomial discriminant to zero to obtain a polynomial which contains both unstable configurations and double points, and (iii) eliminating extraneous roots from algebraic manipulations and double points from this composite polynomial to determine the unstable configurations. The computational procedure is illustrated through numerical examples.