Development of appropriate structured mathematical models for fish population dynamics along with efficient numerical schemes for the solutions of the developed and other models of engineering problems

Abstract

This thesis is mainly concentrated to obtain numerical solutions of real world problems encountered in continuum mechanics and other branches of sciences like development, management and production oriented agricultural sectors. Appropriate mathematical models are well established for all continuum mechanics problems and hence in such instances a faster algorithm or a technique is the only requirement for obtaining numerical solutions. On the other hand, though the dynamic model approach is a widely applied technique to a number of environmental management and sustainability issues like fisheries management problems still needed to be developed. Therefore, primarily the thesis intends to develop appropriate mathematical models for such real problems and then stresses to obtain their numerical solutions. More specifically, we intended firstly to develop appropriate mathematical model in order to calculate fish population. As an outcome, we finally present the model by a system of hyperbolic partial differential equations with linear and nonlinear boundary conditions for the calculation of fish population. Secondly, an appropriate model for mathematical estimation of fish production performances is developed for the calculation of fish sizes in different time span depending on initial sizes. Then, as an important integral component, computer codes in FORTRAN that employs Finite Volume Method are developed for obtaining numerical solutions of such models. Afterward, other codes in MATLAB are developed for analyzing and graphical presentation of computed data. Substantiation of the outcomes of the developed models is then established by comparing the computed results with the experimental data. The versatility and popularity of Finite Element Method (FEM) is well known. The main and important time consuming step in FEM is the formation of all element matrices. Generally, all the elements in global space are transformed into respective contiguous elements in local space by use of isoparametric/ Abstract

subparametric/ superparametric transformation. For such transformations only all the components of element matrices become integrals of rational functions for the popular quadrilateral as well as for the curved triangular finite elements. The Gaussian quadrature schemes, used in most cases for its simplicity cannot evaluate such rational integrals as it can evaluate the integrals of polynomials of degree/ order (2n-1) with n Gaussian points. For the desired accuracy of the evaluations, more and more Gaussian points are used and eventually that increases the computing time. Therefore, it is an important task to make a proper balance between the accuracy and efficiency of evaluations of numerous rational integrals. So, the thesis concentrates to develop the faster technique by reducing steps of various stages of usual FEM solution procedure for obtaining numerical solutions of numerous boundary value problems governed by hyperbolic, elliptic partial differential equations. For doing so, it stresses to present faster closed form formulae needed to form exactly all types of element matrices for solving such two dimensional boundary value problems encountered in the realm of science and engineering. Since, the faces of finite volume are finite elements so all the formulae are applicable in both FEM, FVM methods. Computer codes compatible with the formulations are also developed accordingly. The efficiency and accuracy of the technique is then demonstrated through application of the formulae in order to obtain the solutions of test problems. Thus, in brief the Thesis includes: (1) appropriate mathematical model for the calculation of fish population, (2) appropriate mathematical model for the calculation of fish population performances (size of fishes), (3) computer codes employing suitable numerical methods (FVM) for obtaining best approximations of solutions of the developed models, and (4) computer codes based on the developed technique for exact computing all the element matrices efficiently in order to solve numerous two dimensional boundary value problems. All the relevant concepts, mathematical tools, devised; modified; improved algorithms, other related topics and the gradual development of the Thesis work are elaborately described in 7 (seven) chapters.