The aim of the present work is to introduce solution of special dual integral equations by the orthogonal polynomials. We consider a system of dual integral equations with trigonometric kernels which appear in formulation of the potential distribution of an electrified plate with mixed boundary conditions and convert them to Cauchy type singular integral equations. We use the Chebyshev orthogonal polynomials to construct approximate solution for Cauchy type singular integral equations which will solve the main dual integral equations. Numerical results demonstrate effectiveness of this method.

This paper investigates the optimal portfolio control problem under a stochastic volatility model, whose dynamics are governed by a highly nonlinear Hamilton–Jacobi–Bellman equation. We employ a separable value function and introduce a novel exponential approximation technique to simplify the nonlinear terms of the auxiliary function. The simplified HJB equation is solved numerically using the advanced Fréchet–Newton method, which is known for its rapid convergence properties. We rigorously analyze the numerical outcomes, demonstrating that the iterative sequence converges quickly to the trivial fixed point ($g^{*}=1$) under zero risk and zero excess return conditions. This convergence is mathematically justified through rigorous functional analysis, including the principles of contraction mapping and the Kantorovich theorem, which validate the stability and efficiency of the proposed numerical scheme. The results offer theoretical insight into the behavior of the HJB equation in simplified solution spaces.