Diagonally implicit multistep block methods for solving first order ordinary and fuzzy equations

In this study, two-point diagonally implicit multistep block methods are proposed for solving single first order ordinary and fuzzy differential equations. The methods are based on the diagonally implicit multistep block methods. It approximates two points simultaneously at n 1 y  and n 2 y  in a block along the interval. Subsequently, the methods of order three, four and five are implemented and numerically tested using constant step size. The numerical results show that the two-point diagonally implicit multistep block methods could solve the ordinary differential equations without any difficulty. These methods are also able to reduce the number of steps and execution times even when the number of iterations is being increased. Meanwhile, the first order fuzzy differential equations is interpreted based on Seikkala’s derivative. By including characterization theorem, the fuzzy differential equations can be replaced by the equivalent system of ordinary differential equations. The numerical results show that the two-point diagonally implicit multistep block methods could solve the fuzzy differential equations. The accuracy of the approximate solutions is obtained by means of implementation of the method under the Seikkala’s derivative interpretation. Nevertheless, these methods respectively have the advantage in terms of reducing the number of function evaluations, total steps and execution times. In conclusion, the diagonally implicit multistep block methods are suitable for solving the single first order ordinary and fuzzy differential equations.

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