In this article, we consider a unified generalized version of extended Euler’s Beta function’s integral form a involving Macdonald function in the kernel. Moreover, we establish functional upper and lower bounds for this extended Beta function. Here, we consider the most general case of the four-parameter Macdonald function $K_{\nu +{\displaystyle \frac{1}{2}}}\left(p{t}^{-\lambda }+q{(1-t)}^{-\mu }\right)$ when $\lambda \ne \mu$ in the argument of the kernel. We prove related bounding inequalities, simultaneously complementing the recent results by Parmar and Pogány in which the extended Beta function case $\lambda =\mu$ is resolved. The main mathematical tools are integral representations and fixed-point iterations that are used for obtaining the stationary points of the argument of the Macdonald kernel function $K_{\nu +{\displaystyle \frac{1}{2}}}$.

We obtain improved derivative estimates for the solution of the semilinear singularly perturbed reaction-diffusion problem in one dimension. This enables us to modify the transition points between the fine and coarse parts of the Shishkin discretization mesh. We prove that the numerical solution, obtained by using the central finite-difference scheme on the modified mesh, retains the same order of convergence uniform in the perturbation parameter as on the standard Shishkin mesh. However, the modified mesh may be denser in the layers than the standard one, and, when this is the case, numerical results show an improvement in the accuracy of the computed solution.