This paper considers the first boundary value problem (Dirichlet problem) for a semilinear elliptic equation that serves as a mathematical model of a thermochemical process with Arrhenius kinetics. The nonlinearity in the equation is exponential with saturation. By means of Green’s function, the considered boundary value problem is reduced to an equivalent Hammerstein integral equation. This equation is used to define a generalized solution and is analyzed using methods of nonlinear operator theory in semiordered Banach spaces. In particular, the integral equation is represented as an equation with a nonlinear operator acting in the space of continuous functions partially ordered by the cone of nonnegative functions. The properties of the nonlinear operator are studied, including positivity, monotonicity, Lipschitz continuity, and the existence of an invariant cone segment. For the numerical analysis of the integral equation (and hence the original boundary value problem), an iterative scheme based on the method of two-sided approximations is proposed, where the endpoints of the invariant conical segment are chosen as initial approximations. Conditions for the convergence of the iterations and for the existence of a positive solution to the boundary value problem are established. In addition, a two-sided a priori estimate for the solution is obtained. Computational experiments are carried out for various parameter values in the unit disk and the unit ball. Conclusions are drawn regarding the efficiency of the proposed numerical method and the physical properties of the solution depending on the parameters. An advantage of the two-sided iteration method is the availability of a guaranteed a posteriori error estimate for the approximate solution, as well as a convenient stopping criterion for the computational process.

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