Diagonally implicit two point block backward differentiation formulas for solving stiff ordinary differential equations and fuzzy differential equations - Mohd Zawawi, Iskandar Shah (2014)
Our main contribution in the thesis is the development of a new block method which is called diagonally implicit two point block backward differentiation formulas of order two (DI2BBDF(2)), order three (DI2BBDF(3)) and order four (DI2BBDF(4)) for solving stiff ordinary differential equations (ODEs) and fuzzy differential equations (FDEs). This method is constructed to compute multiple approximations concurrently in a block using various back values. The performance of the method is compared with existing methods. Furthermore, the convergence and stability properties of the method are investigated. The strategy of choosing suitable step size is also discussed. This thesis also explored the numerical solution of first order FDEs. The fully implicit two point block backward differentiation formulas of order three (FI2BBDF(3)) is reviewed and modified in fuzzy version for solving fuzzy initial value problems (FIVPs) under a new interpretation of Hukuhara Differentiability Theorem (HDT). Based on HDT, the exact and approximated solutions for two cases are compared to investigate the accuracy of the method. Finally, the derived method is modified in fuzzy version for solving FIVPs under HDT. The efficiency of the method is compared with several existing methods. In conclusion, the proposed method can be an alternative method for solving stiff ODEs and FDEs.