New approximation methods based on fuzzy transform for solving ordinary differential equations

Real world probelms in science and engineering are modelled by using differential equations. In many cases, the differential equations cannot be solved analytically so that numerical methods are required. As time goes on, researchers realized that fuzzy approaches are particularly that has been proposed in the literature is the fuzzy transform (FzT). There has been a growing interest in investigating the properties of fuzzy partitions. In this research, new representations of basic functions are proposed. This is achieved by introducing new generalized uniform fuzzy partitions called power of the triangular and raised cosine generalied uniform fuzzy partitions. The main properties of the new generalized uniform fuzzy partitions are proposed. Further, the simpler from of FzT is given alongside with some of its fundamental results. New theorems and lemmas are proposed and proved mathematically. Then, the new generalized uniform fuzzy partitions are used in three approximation methods based on FzT to solve Cauchy problems. The first approximation method used Trapezoidal rule with FzT and new iteration method (NIM). The results proved that the first approximation method convereged to the exact solution. The second approximation method used Adams Moulton method with FzT and NIM while the thrid approximation method used Adams Moulton method with FzT and NIM. From the numerical results, all the proposed fuzzy approximation methods outperform the classical Trapezoidal rule and classical Adams Moulton method. Further, it is also observed that the proposed fuzzy approximation methods are more accurate in comparison with the existing fuzzy approximation methods. This result is an important improvement to the previous results for Cauchy problems. Furthermore, two approximation methods are used in the fuzzy partitions to solve system of differential equations based on FzT and a one step method. The first approximation method used Euler method with FzT and the second approximation method used Trapezoidal rule with FzT. From the numerical results, it is observed that both fuzzy approximation methods yield more accurate results in comparison with the classical Euler method and classical Trapezoidal rule. This result is an important improvement to the previous results for solving system of differential equations. The discussion continued in last part with more advanced techniques, In the last part, the new generalized uniform fuzzy partitions are used in three approximation methods based on FzT to solve system of differential equations. In accordance with the three approximation methods for Cauchy problem, Trapezoidal rule and Adams Moulton method are improved using FzT and NIM. From the numerical results, it is observed that the new fuzzy approximation methods yield more accurate results in comparison with the classical Trapezoidal rule and classical Adams Moulton method. Hence, the new fuzzy approximation methods provide alternative techniques for solving differential equations with better results.