Higher derivative block methods of generalised steplength for the direct solution of fuzzy ordinary differential equations - Hussain, Kashif (2023)
Differential equations are useful in modelling various concepts and scenarios. However, when considering the initial or boundary conditions for these differential equation models, the use of fuzzy numbers is more realistic and flexible, as the parameters may vary within a certain range. Such situations are referred as unpredictable circumstances which introduce the concept of uncertainty. Thus, fuzzy derivatives and fuzzy differential equations (FDEs) are used to deal with these situations. Numerical methods are used to provide approximate solutions to differential equations, such as FDEs, in the absence of exact solutions. Conventionally, higher-order fuzzy ordinary differential equations (FODEs) are solved by first reducing the equations to their equivalent systems of first-order FODEs. Then, suitable existing numerical methods for first-order FODEs are employed to solve the resulting systems. This approach results in having more equations and thus increases the computational burden which jeopardises the accuracy of the solution. For this reason, several existing studies have developed numerical methods for the direct solution of FODEs. However, the results obtained possess accuracy that could be improved. Other issues encountered in existing studies that consider the numerical solution of FODEs include the methods being non self-starting, not solving both fuzzy initial value problems (FIVPs) and fuzzy boundary value problems (FBVPs), and the absence of a generalised form for the developed numerical methods. Therefore, to overcome these problems, this study developed more accurate higher derivative self -starting block methods with generalised steplength for the direct numerical solution of FODEs. A linear block approach using Taylor series expansion is adopted for the derivation of the block methods with the presence of two higher derivatives terms. Investigations on the order, error constant, zero-stability, consistency, convergence, and region of absolute stability of the new block methods with the presence of two higher derivatives were also conducted, and the developed new block methods satisfied all convergence properties in the fuzzy form successfully. To investigate the accuracy, the developed block methods were employed to solve both FIVPs and FBVPs considered in the literature. The numerical results confirmed the superiority of the new methods over the existing methods when solving the same test problems in terms of absolute errors. In conclusion, this study has successfully developed self -starting and efficient block methods in solving FODEs directly.