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The Grad–Shafranov plasma equilibrium equation was originally solved analytically in toroidal geometry, which fitted the geometric shape of the first Tokamaks. The poloidal surface of the Tokamak has evolved over the years from a circular to a D-shaped ellipse. The natural geometry that describes such a shape is the prolate elliptical one, i.e., the cap-cyclide coordinate system. When written in this geometry, the Grad–Shafranov equation can be solved in terms of the general Heun function. In this paper, we obtain the complete analytical solution of the Grad–Shafranov equation in terms of the general Heun functions and compare the result with the limiting case of the standard toroidal geometry written in terms of the Fock functions. [ABSTRACT FROM AUTHOR]

In this paper, we construct an analytical solution of the coupled Burgers' equation, using the homotopy analysis method, which is a semi-analytical method, the approximate solution obtained by this method is convergent for different values of the convergence control parameter ℏ, the optimal value of ℏ corresponding with the minimum error to be determined by the residual. The results obtained by the present method are compared with other obtained solutions by different numerical methods. [ABSTRACT FROM AUTHOR]

In the sense of a conformable fractional operator, we consider a generalized fractional–stochastic nonlinear wave equation (GFSNWE). This equation may be used to depict several nonlinear physical phenomena occurring in a liquid containing gas bubbles. The analytical solutions of the GFSNWE are obtained by using the F-expansion and the Jacobi elliptic function methods with the Riccati equation. Due to the presence of noise and the conformable derivative, some solutions that were achieved are shown together with their physical interpretations. [ABSTRACT FROM AUTHOR]

The stochastic fractional-space Korteweg–de Vries equation (SFSKdVE) in the sense of the M-truncated derivative is examined in this article. In the Itô sense, the SFSKdVE is forced by multiplicative white noise. To produce new trigonometric, hyperbolic, rational, and elliptic stochastic fractional solutions, the tanh–coth and Jacobi elliptic function methods are used. The obtained solutions are useful in interpreting certain fascinating physical phenomena because the KdV equation is essential for understanding the behavior of waves in shallow water. To demonstrate how the multiplicative noise and the M-truncated derivative impact the precise solutions of the SFSKdVE, different 3D and 2D graphical representations are plotted. [ABSTRACT FROM AUTHOR]

Here, we analyze the (2+1)-dimensional stochastic modified Kordeweg–de Vries (SmKdV) equation perturbed by multiplicative white noise in the Stratonovich sense. We apply the mapping method to obtain new trigonometric, elliptic, and rational stochastic fractional solutions. Because of the importance of the KdV equation in characterizing the behavior of waves in shallow water, the obtained solutions are beneficial in interpreting certain fascinating physical phenomena. We plot our figures in MATLAB and show several 3D and 2D graphical representations to show how the multiplicative white noise affects the solutions of the SmKdV. We show that the white noise around zero stabilizes SmKdV solutions. [ABSTRACT FROM AUTHOR]

We present some solutions of the three-dimensional Laplace equation in terms of linear combinations of generalized hyperogeometric functions in prolate elliptic geometry, which simulates the current tokamak shapes. Such solutions are valid for particular parameter values. The derived solutions are compared with the solutions obtained in the standard toroidal geometry. [ABSTRACT FROM AUTHOR]