APPROXIMATE SOLUTIONS FOR SEVERAL CLASSES OF FUZZY DIFFERENTIAL EQUATIONS USING RESIDUAL POWER SERIES METHOD
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Fuzzy differential equations provide a crucial tool for modelling numerous phenomena and the uncertainties that potentially arise, with various applications across physics, applied sciences and engineering. Given that it is generally problematic to obtain accurate solutions for such fuzzy differential equations, this means that reliable and effective analytical methods are necessary to obtain required solutions. In this thesis, the residual power series (RPS) method is considered for solving several classes of fuzzy differential equations, in addition to uncertain initial guess data under the concept of strongly generalized differentiability. The RPS provides a systematic scheme based on Taylor series expansion and minimisation of residual error functions, so as to obtain the coefficient values of the power series based on the given initial data of triangular fuzzy numbers. The solution methodology is reliant on fuzzy differential equations’ conversion into a system of ordinary differential equations, utilising the parametric form with regard to its -level representations. Subsequently, the equivalent classical systems are resolved by applying the RPS algorithm. This thesis tackles several fuzzy topics of differential equations, including fuzzy Fredholm integro-differential equations, fuzzy Volterra integro-differential equations, quadratic fuzzy differential equations, as well as cubic fuzzy differential equations. The fuzzy quadratic Riccati differential equations are proposed and solved through adopting the RPS algorithm. The approximate series solutions are provided within a radius appropriate to the requisite domain. Additionally, numerical applications are introduced as a means of clarifying the RPS algorithm’s compatibility and reliability. Following this, the method is implemented to establish a fuzzy approximate solution for nonlinear fuzzy Duffing oscillator problem, which is a cubic fuzzy differential equation. The method’s effect and capacity are verified through testing it on some applications. Parameter ’s level effect are presented graphically and quantitatively, expressing a strong agreement between the classic and fuzzy approximate solutions. Moreover, fuzzy Fredholm integro-differential equations are assessed as being first order in terms of their linear and nonlinear characteristics. Similarly, the first order linear fuzzy Volterra integro-differential equation is solved through applying this proposed method. Afterwards, the obtained results are numerically compared with the classical solutions and those obtained via other existing methods, indicating that the proposed method is a suitable and remarkably powerful scheme for solving the varied linear and nonlinear uncertain natural problems arising within the applied physics and engineering fields. Furthermore, the RPS method has been formulated as an effective analytical-numerical approach to identifying unknown coefficients of power series solutions for first and second-order fuzzy differential equations, which poses numerous advantages. First, the RPS method is an advantageous tool for creating power series solutions for both linear and nonlinear equations without resorting to linearization, perturbation or discretization. Second, it may be applied directly for the specific problem, through choosing an appropriate value for the initial approximation. Third, the RPS method only has small computational requirements while simultaneously providing high accuracy and requiring less time. Finally, the exact solution will be derived whenever the solution is polynomial.