Exact Solutions for a Class of Wick-Type Stochastic (3+1)-Dimensional Modified Benjamin–Bona–Mahony Equations

Abstract

In this paper, we investigate the Wick-type stochastic (3+1)-dimensional modified Benjamin–Bona–Mahony (BBM) equations. We present a generalised version of the modified tanh–coth method. Using the generalised, modified tanh–coth method, white noise theory, and Hermite transform, we produce a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations. This set includes solutions of exponential, hyperbolic, and trigonometric types. With the help of inverse Hermite transform, we obtained stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Eventually, by application example, we show how the stochastic solutions can be given as white noise functional solutions.

Hypercomplex systems and non-Gaussian stochastic solutions of χ-Wick-type (3+1)-dimensional modified Benjamin-Bona-Mahony equation

“…1, with variable coefficients. Through white noise functional analysis [16], Ghany [17,18], Ghany and Hyder [19][20][21][22], Ghany et al [23,24], Ghany and Zakarya [25,26], Hyder and Zakarya [27], Agarwal et al [28], Zakarya et al [29], Agarwal et al [30] studied this model of white noise functional solutions for some non-linear stochastic partial differential equations (SPDE) more intensively. In addition, Okb El Bab et al, [1] and Zakarya [3] explored some important topics related to the construction of non-Gaussian white noise analysis using the hyper-complex systems theory and some applications.…”

“…Dobroshin and Minlos [33] had a comprehensive study of this particular topic in both mathematical physics and probability theory. Currently, the Wick product provides a useful concept for various applications, for example in the study of stochastic ordinary and PDE, see [26][27][28][29][30]. In this section, we present a new product and definition of the so-called χ-Wick product and χ-Hermite transform in the space D χ -q , respectively, with regard to the non-Gaussian probability measure λ.…”

Fractional traveling wave solutions of the (2 + 1)‐dimensional fractional complex Ginzburg–Landau equation via two methods

“…Exact solutions of nonlinear models play an important part in the explanation and description of engineering and physical issues, including optics, fluid dynamics, electromagnetism, plasma physics, and other fields including nonlinear wave propagation phenomenon 1–3 . Fractional nonlinear models (FNMs) were also widely utilized in different domains of engineering, 4,5 physics, 6–9 biology, 10–14 and mathematics 15,16 . Because many systems exhibit post effects or memory, fractional derivatives are a better explanation, 17 and thus, FNMs can be extended to model a variety of complex phenomena 18–20 …”

More on Hölder’s Inequality and It’s Reverse via the Diamond-Alpha Integral

“…For the development of dynamic inequalities on a time scale calculus, we refer the reader to the articles in [19][20][21][22][23][24][25][26][27][28]. Although there are many results for time scale calculus in the sense of delta and nabla derivative, there is not much done for diamond-α derivative.…”

“…In Theorem 4, if we make the substitution u(ϑ) = M = m and s − r = 1 with λ = 2, then the equality in(24) holds.…”