A meshfree approach for the rennet-induced coagulation equation: Spline based multistage Bernstein collocation method and its convergence analysis

The initial phases of milk coagulation for cheese manufacturing can be tracked by an integro-differential equation known as a population balance equation. In this article, a new analytical approach using multistage Bernstein polynomials is presented to solve a rennet-induced coagulation equation for the first time. The existence of the solution and convergence analysis of the proposed approach are discussed in detail to support the mathematical formulation. Our main interest is in computing the integral moments, such as the number and total volume/mass of casein micelles over time. These moments are evaluated by approximating them with the linear combinations of Bernstein polynomials that involve unknown coefficients. Furthermore, the unknown coefficients are determined by selecting an appropriate number of collocation points, based on the considered time span of the process. To test the accuracy and efficiency of the new approach, the new analytical solutions for the integral moments are obtained for constant, sum and product coagulation kernels and results are verified by comparing with the existing finite volume scheme and Picard's method.