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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

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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.

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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