A uniformly convergent numerical scheme is developed for solving a singularly perturbed parabolic turning point problem. The properties of continuous solutions and the bounds of the derivatives are discussed. Due to the presence of a small parameter as a multiple of the diffusion coefficient, it causes computational difficulty when applying classical numerical methods. As a result, the scheme is formulated using the Crank-Nicolson method in the temporal discretization and an exponentially fitted finite difference method in the space on a uniform mesh. The existence of a unique discrete solution is guaranteed by the comparison principle. The stability and convergence analysis of the method are investigated. Two numerical examples are considered to validate the applicability of the scheme. The numerical results are displayed in tables and graphs to support the theoretical findings. The scheme converges uniformly with order one in space and two in time.

In this paper, we investigate the canonical property of solutions of a system of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm-Liouville equation with a turning point. Using the asymptotic estimates for a special fundamental system of solutions of Sturm-Liouville equation, we study the infinite product representation of solutions of the system and investigate the uniqueness of the solution for the dual equations of the Sturm- Liouville equation. Then, we transform the Sturm-Liouville equation with a turning point to the equation with a singularity, and study the asymptotic behavior of its solutions. Such representations are relevant to the inverse spectral problem.

In this paper, we apply the Olver’s solution of second order differential equation with the two turning points case. Note that the weight function in this equation has two zeros in the domain, the so called "turning points". As a basic result we derive the distribution of the eigenvalues with the Neumann boundary conditions.

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