Direct block backward differentiation formulas for solving second order stiff boundary value problem

In this thesis, the direct method of Block Backward Differentiation Formula (BBDF) for solving two point boundary value problems (BVPs) directly was studied. The shooting technique will be implemented using constant step size. In order to overcome the numerical instabilities due to round off or truncation errors that occur in solving BVPs, the BBDF method will be adapted with multiple shooting techniques. Newton-Raphson method is also considered as a procedure for solving the second order BVPs. Existing strategy for solving BVPs, is by reducing them to a system of first order ordinary differential equations (ODEs). This approach is well established but obviously it will enlarge the problem into a system of first order equations. However, the BBDF method in this thesis solves BVPs directly without reducing them to their first order differential equations. Besides that, the BBDF method can produce two approximate solutions at two points in each step. Furthermore, the BBDF method allows the differentiation coefficients to be stored and thus reduces the computational cost. Another main focus in this thesis is to solve stiff BVPs, where more computational efforts are required to evaluate the Jacobian and solving the linear systems. Furthermore, stiff BVPs are difficult to solve due to the restriction on the step size of many numerical methods, except those with A-stability properties. Therefore, the BBDF method in this thesis will be used to solve stiff BVPs directly. The source codes are written in C language and executed using MATLAB. Some numerical examples are given to illustrate the efficiency of the method Numerical results showed that the BBDF method manages to give acceptable results in terms of maximum error, number of iterations and execution time. In conclusion, the proposed BBDF method in this thesis is suitable for solving directly the second order BVPs.

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