New Approximation Methods Based on Fuzzy Transform for Solving SODEs: I

Abstract

In this paper, new approximation methods for solving systems of ordinary differential equations (SODEs) by fuzzy transform (FzT) are introduced and discussed. In particular, we propose two modified numerical schemes to solve SODEs where the technique of FzT is combined with one-stage and two-stage numerical methods. Moreover, the error analysis of the new approximation methods is discussed. Finally, numerical examples of the proposed approach are confirmed, and applications are presented.

On the Numerical Solution of Ordinary, Interval and Fuzzy Differential Equations by Use of F-Transform

“…(Fuzzy-valued problem FDE1): For eleven values of α ∈ i−1 10 |i = 1, ..., 10 , the table contains the interval level-wise initial condition (column 2), the final interval value with Meth = 1 (column 3) and the final interval value with Meth = 2. where the quantity MSE (mean squared error of approximate solution and the exact one) is computed. Ex4 is Example 1 in [14] and Ex5 is Example 3 in [14]. The best MSE quantities obtained by [14] and by ODE-FT are reported in Table 6: Table 6.…”

“…In particular, it is shown that by using the inverse F-transform, it is possible to obtain good approximations of the solution x(t). The methods that use the F-transform are (computationally) superior with respect to other ones such as the second-order Runge-Kutta algorithm or basic multi-step algorithms (see [12][13][14][15][16]). In the final section of this paper, we will present some comments and a preliminary comparative valuation of the proposed methods.…”

New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II

“…Consider the problem of Lotka-Volterra-prey-predator model (36). We take α (t) = 4 + tan (t) , β (t) = exp(2t), γ (t) = −2, δ (t) = cos(t), x(0) = −4 and y(0) = 4.…”

“…The motivation of the proposed study comes from [16,36,37]. In [16], new fuzzy numerical methods to solve the Cauchy problem was considered and the authors showed that the error can be reduced by FzT and NIM with respect to new generalized uniform fuzzy partitions, namely power of the triangular and raised cosine generalized uniform fuzzy partitions, where generating functions are normal (see also [37] for another approach).…”

“…In [16], new fuzzy numerical methods to solve the Cauchy problem was considered and the authors showed that the error can be reduced by FzT and NIM with respect to new generalized uniform fuzzy partitions, namely power of the triangular and raised cosine generalized uniform fuzzy partitions, where generating functions are normal (see also [37] for another approach). In addition, two basic approximation methods, modified Euler method and Trapezoidal rule, with help from FzT for solving SODEs are analyzed in detail by [36]. For this purpose, more generally, new generalized uniform fuzzy partitions are proposed in this study, where a generating function is not normal.…”