GENERALIZED MULTI-BLOCK BOUNDARY VALUES METHODS FOR DIFFERENTIAL AND DIFFERENTIAL ALGEBRAIC EQUATIONS

Abstract

Interest in boundary value methods has emerged as an alternative means for dealing with high computationally cost occasioned by stiffness in differential equations which arises from practical modeling problems. This study introduces very large integration block methods which offer boundary value methods properties such as integration efficiency. The method proposed in the study are known as multi-block boundary value methods. The proposed MB2V Ms are based on the linear multi-block methods in the sense of the conventional boundary value methods. The root distribution of the associated stability polynomial of the new class of methods are determined using the Wiener-Hopf factorization of a matrix polynomial for the reason of their correct implementation. Families of the MB2V Ms includes: the multi-block generalized backward differentiation formulas, multi-block generalized Adams methods, multi-block extended trapezoidal rule of second kind, un-symmetric multi-block extended trapezoidal rule of second kind, multi-block top order methods, multi-block generalized extended backward differentiation formulas, while families of second derivative multi-block boundary value methods were considered as well. Application of the MB2V Ms in the integration of stiff system of initial value problems in ordinary differential equations and differential algebraic equations shows that the methods are suitable. Families of the MB2V Ms have been implemented in block mode using fixed step size and compared favourably with the results of some known methods in the literature