Solving directly special third and fourth order ordinary differential equations using linear multistep and explicit Obrechkoff methods

This study focuses mainly on developing linear multistep methods which can directly solve special third and fourth order ordinary differential equations (ODEs). We constructed and derived the new explicit and implicit linear multistep methods for different stepnumbers based on Taylor’s series expansion. The study in the thesis consists of three parts. The first part of the thesis described the derivation of explicit and implicit multistep methods with step number k equals to three, five and six for directly solving special third order ODEs. The stability of the methods is also investigated. The numerical results revealed that the new methods are more efficient as compared to the existing methods. The second part of the thesis focused on the derivation of explicit and implicit multistep methods with step number k equals to four and five for directly solving special fourth order ODEs. The zero-stability and absolute stability of the new methods are also given. 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. Finally, the last part of the thesis concerned with the construction of explicit multistep method with extra derivative known as Obrechkoff methods for directly solving special third order ODEs. Stability properties of the methods are also presented. Numerical results show that new methods are more efficient than the existing methods. As a whole, the new proposed methods for directly solving special third and fourth order ordinary differential equations have been presented. The illustrative examples demonstrate the superiority of the new linear multistep and Obrechkoff methods over existing numerical methods in the scientific literature.

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