Solving higher order delay differential equations with boundary conditions using multistep block method

In this thesis, we derived two numerical methods called two point diagonally multistep block method order four and order five with the approach of predictorcorrector technique to solve higher order delay differential equations (DDEs) with boundary conditions. Shooting technique by using the Newton’s like method is implemented to solve the boundary value problems (BVPs). This thesis begins with solving second order DDEs with constant, pantograph and time dependent delay type by using both methods. Then, those methods are extended to solve third order DDEs with constant and pantograph delay type. The approach used to solve constant delay type is by taking the previously calculated solutions at the delay terms while for pantograph and time dependent delay types, the approaches are by using the Lagrange interpolation to approximate the solutions at the delay terms. The derivatives present in the problems at the delay terms will be approximated by using the finite difference method. The analysis of both methods in terms of order, local truncation error and stability are also investigated. Two stability test equations are used to analyze the stability regions of the block methods. Several numerical problems are illustrated to solve by using C programming. The accuracy of the methods in terms of maximum and average errors along with the total function calls, total iteration steps, total guessing numbers for shooting technique are discussed and compared with the previous methods. In conclusion, the higher order DDEs with boundary conditions can be solved by using the proposed block methods based on the analysis of the methods and their numerical results.

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