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Let p be a real density function defined on a compact subset of R m , and let E(f , p) = pf dω be the expectation of f with respect to the density function p. In this paper, we define a one-parameter extensionof a positive continuous function f defined on . By means of this extension, a two-parameter mean V r,s (f , p), called the Dresher variance mean, is then defined. Their properties are then discussed. In particular, we establish a Dresher variance mean inequality min t∈ {f (t)} ≤ V r,s (f , p) ≤ max t∈ {f (t)}, that is to say, the Dresher variance mean V r,s (f , p) is a true mean of f . We also establish a Dresher-type inequality V r,s (f , p) ≥ V r * ,s * (f , p) under appropriate conditions on r, s, r * , s * ; and finally, acan be compared with E(f , p). We are also able to illustrate the uses of these results in space science. MSC: 26D15; 26E60; 62J10

In this paper, the dynamical behavior of a time-space fractional Phi-4 equation is investigated by utilizing the bifurcation method of a planar dynamical system. Under the given parameter conditions, phase portraits and bifurcations are obtained with the help of the mathematical software Maple. Moreover, some new exact traveling wave solutions are obtained, such as Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, kink solitary wave solutions, and periodic wave solutions.

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The fractional perturbed Gerdjikov–Ivanov (pGI) equation plays a momentous role in nonlinear fiber optics, especially in the application of photonic crystal fibers. Constructing traveling wave solutions to this equation is a very challenging task in physics and mathematics. In the current article, our main purpose is to give the classifications of traveling wave solutions of the fractional pGI equation. These results can help physicists to further explain the complex fractional pGI equation.

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In this paper, the bifurcation and new exact solutions for the ($2+1$ 2 + 1 )-dimensional conformable time-fractional Zoomeron equation are investigated by utilizing two reliable methods, which are generalized $(G'/G)$ ( G ′ / G ) -expansion method and the integral bifurcation method. The exact solutions of the ($2+1$ 2 + 1 )-dimensional conformable time-fractional Zoomeron equation are obtained by utilizing the generalized $(G'/G)$ ( G ′ / G ) -expansion method, these solutions are classified as hyperbolic function solutions, trigonometric function solutions, and rational function solutions. Giving different parameter conditions, many integral bifurcations, phase portraits, and traveling wave solutions for the equation are obtained via the integral bifurcation method. Graphical representations of different kinds of the exact solutions reveal that the two methods are of significance for constructing the exact solutions of fractional partial differential equation.

In this paper, the classification of all single traveling wave solutions to generalized fractional Gardner equations is presented by utilizing the complete discrimination system method. Under the fractional traveling wave transformation, generalized fractional Gardner equations can be reduced to an ordinary differential equations. All possible exact traveling wave solutions are given through the complete discrimination system of the fourth-order polynomial. Moreover, graphical representations of different kinds of the exact solutions reveal that the method is of significance for searching the exact solutions to generalized fractional Gardner equations.