Second derivative block methods for solving first and higher order ordinary differential equations

Traditionally, higher order ordinary differential equations (ODEs) are solved by reducing them to an equivalent system of first order ODEs. However, it is more cost effective if they can be solved directly by numerical methods. Block methods approximate the solutions of the ODEs at more than one point at one time step, hence faster solutions can be obtained. It is rather well-known too that a more accurate numerical results can be obtained by having extra derivatives in the method. Based on these arguments, we are focused on developing block methods with extra derivatives for solving first, second and third ODEs. The study in the thesis consists of three parts. The first part of the thesis described the derivation of two and three point implicit and semi implicit block methods with second derivative for solving first order ODEs. Absolute stability for both implicit and semi implicit second derivative block methods are also presented. Numerical results clearly show that the new proposed methods are more efficient in terms of accuracy and computational time when compared with well-known existing methods. The second part of the thesis is focused on the derivation of two and three point implicit and semi implicit second derivative block methods for directly solving second order ODEs. The zero-stability of the new methods are also given. The numerical results revealed that the new methods are more accurate as compared to the existing methods and it is also illustrated that the new second derivative block methods require less computational time when solving second order ODEs. Finally, the last part of the thesis concerned with the construction of two and three point implicit and semi implicit second derivative block multistep methods for directly solving third order ODEs. The zero-stability for the new methods are also presented. Numerical results show that new methods are more efficient than the existing methods. In conclusion, accurate and required less computational time have potential to be a good tools for solving first, second and third order ODEs respectively.

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