In this article, Otway provides a solution to the closed Dirichlet problem which is a mixed eliptic-hyperbolic equation. This type of equations are encountered in electromagnetic wave propagation in cold plasmas.
The equation in the model for electromagnetic wave propagation in zero-temperature plasma is:
(x-y^2)u_{xx}+u_{yy}+\kappa u_{x}=0
where u(x,y) is twice-continuously differentiable function. This is a homogeneous closed Dirichlet problem with D-star-shaped domains. To determine the boundary conditions, a geometric or physical analogy is considered to solve the equation. The boundary arcs are good approximations to produce an appropriate vector field. Although this approach is introduced by Lupo and Payne, the results of this study generalize the starlike boundary conditions to elliptic-hyperbolic boundary value problems. Before this study, the closed Dirichlet problems are considered as ill-posed. However, the author describes a novel method to solve the problem by presenting a unique weak solution with suitable weight function. The weight function is determined as y=x^2 and it reduces to y=0 when uniqueness is eliminated. Physically, it refers to a heating point in the plasma satisfying the equation itself.
The unstable heating on plasmas creates anisotropy for electromagnetic waves. The field potential in inhomogeneous regions can be represented as Equation 1. By reducing the number of hybrid waves in the plasma, an unstable heating solution can be obtained with kappa=0. The function domain Omega is defined as open, bounded and connected. The boundary condition is decided as:
u(x,y)=0 for all (x,y) in Omega.
In the study, it is shown that there is a solution to a closed Dirichlet problem by proving the existence of L^2 solutions where L is differential operator, such that:
Lu=[K(x,y)u_x]_x+u_{yy}=f(x,y)
where K(x,y)=x-y^2. The advantage of this approach is that a homogeneous equation with singularity is converted into an inhomogeneous equation by avoiding trivial solutions with known f(x,y).
For the weak solution, the problem is simplified by ignoring kappa. The general solution can be obtained by defining u=(u_1(x,y),u_2(x,y)). For this strong solution, K(x,y)=x-sigma(y) is chosen.
(Lu)_1=[x-\sigma(y)]u_{1x}+u_{2y}+\kappa_1u_1+\kappa_2u_2 and (Lu)_2=u_{1y}-u_{2x}
where kappa_1 and kappa_2 are constants. The reduced forms of these equations imply Cinquini-Cibrario equation which presents applicable models for atmospheric and space plasmas.
The article is well-explained in terms of the clarity of the mathematical methods; however, the author failed to recognize the physical phenomena, described here as the wave propagation in the plasma. The article could include precisely the temperature level interested and the pressure dependence of plasma waves. The derivation is conducted at zero temperature by neglecting fluid properties. The Maxwell equations are derived for the electric displacement vector. Still, those who are interested in solving elliptic-hyperbolic equations would benefit from this article by approaching to the closed Dirichlet problems with different boundary conditions.
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References:
Otway, Thomas H.,2010, Unique solutions to boundary value problems in the cold plasma model, SIAM Journal on Applied Mathematics, 42(6): 3045-3053.
Lupo, D. and Payne, K.R., 2003. Critical exponents for semilinear equations of mixed elliptic‐hyperbolic and degenerate types. Communications on pure and applied mathematics, 56(3), pp.403-424.