APPROXIMATE SOLUTIONS FOR SEVERAL CLASSES OF FUZZY DIFFERENTIAL EQUATIONS USING RESIDUAL POWER SERIES METHOD
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Abstract
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.