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Abstract
Real life models that are characterized by randomness are best described using Stochastic Ordinary Differential Equations (SODEs). Most SODEs that satisfy existence and uniqueness theorem are often insoluble via the use of analytic methods. Numericals solution are derived and the complexity of generating approximate solutions to SODEs are heightened by the presence of the phenomenon called stiffness. Hence, A-stable numerical methods are desired. This
is a strigent requirement that can only be met by implicit methods.
Two families of A-stable numerical methods for numerical approximation of SODEs are derived using Ito Taylor, Taylor’s series and undetermined coefficients methods. The stability analysis of both families of methods are established using the Boundary locus method.
Families of methods developed are A-stable for 𝑘 ≤ 12. Mean-square stable and strong stable for order 𝑝 = 1. The Numerical implementation generated on the standard test problems in the literature shows that the numerical solution to methods developed are in most cases better when compared to numerical solution generated by existing methods in the literature designed for stiff SODEs. The numerical solutions are also compared with exact solution where they are
available. The numerical solution mimic the exact solution, hence the proposed methods are well suited for the treatment of stiff SODEs.
is a strigent requirement that can only be met by implicit methods.
Two families of A-stable numerical methods for numerical approximation of SODEs are derived using Ito Taylor, Taylor’s series and undetermined coefficients methods. The stability analysis of both families of methods are established using the Boundary locus method.
Families of methods developed are A-stable for 𝑘 ≤ 12. Mean-square stable and strong stable for order 𝑝 = 1. The Numerical implementation generated on the standard test problems in the literature shows that the numerical solution to methods developed are in most cases better when compared to numerical solution generated by existing methods in the literature designed for stiff SODEs. The numerical solutions are also compared with exact solution where they are
available. The numerical solution mimic the exact solution, hence the proposed methods are well suited for the treatment of stiff SODEs.
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