چكيده به لاتين
This thesis discusses the existence of a unique solution and the stability of functional equations. Considering different fractional systems, including fractional differential integral equations as well as fractional integral equations, we first checked the existence of a unique solution for these equations, then we checked the stability of these equations. The stability of fractional equations leads to finding the best approximation for these equations. Our method to check the stability is the Cadariu Radu method and Radu Mihet method, which is derived from the alternative fixed point method (Diaz and Margulies theorem).
Checking the existence of the solution of fractional equations is done in different ways. In all the proposed methods, we consider a class of special controller functions. These controllers prevent system instability. We have considered the nonhomogeneous fractional delay oscillation equation with order sigma and introduced write control functions. We have used the Cadariu Radu method to prove the stability of this equation. and considering the Gauss Hypergeometric function as a control function, we have investigated the concept of Hayers Ulam Rassias Kummer stability using the Cadariu-Radu method for Volterra fractional integral-differential and Volterra fractional integral equations. Also, in this treatise, we have investigated an integral boundary condition problem. We have shown that there is a unique solution for the nonlinear fractional differential integral equation with integral boundary conditions according to the Babenko strategy and the multivariate Mittag-Leffler function. All results, including the existence problem, are proved using the Banach contraction principle and the Lari-Schuder fixed point theorem. At the end of all the results we have obtained, we present their application using several practical examples. In recent years, there has been an increasing interest in the stability of functional equations in different spaces. The concepts of classical mathematics have been developed by substituting fuzzy sets for classical sets. Addressing the problem of stability in fuzzy norm spaces is a new problem based on which a system in fuzzy conditions is approximated with a control function and the stability of the system in fuzzy conditions is investigated. In this space, a new set called matrix value sets is defined and the space defined on these matrix value sets is used. In matrix value spaces, controllers are also defined as matrix values. We use the Radu-Mihet method derived from the alternative fixed point method in K-norm fuzzy space to obtain approximations for fractional Volterra equations and adaptive fractional differential equations. Considering the matrix value H-Fox function as the control function, the existence of a unique solution, and the Hayers-Ulam-H-Fox stability, we have investigated a Conformable fractional differential equation with a constant coefficient. Another method used in this treatise is the Picard method, based on which we have investigated the existence of a unique solution using Banach's fixed point theorem. We have considered a group of fractional order differential equations and examined two aspects of these equations. First, we investigated the existence of a unique solution, and then, using a new class of control functions, we determined the stability of Gauss Hypergeometrics by considering Chebyshev and Bielecki norms. Considering the importance of using non-linear equations in investigating many natural phenomena, in this treatise, the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation and we have obtained the one-wave, two-wave, and three-wave soliton solutions of this equation using the multiple exponential function method. In the following, the stochastic time-fractional Sine–Gordon equation is studied and the new clique polynomial method is used to obtain the solutions of this equation. In this method, the clique polynomial is considered as a basis function for operational matrices. By converting the two-dimensional ST-FS-G equation into algebraic equations, we have obtained solutions that have been shown by calculating the error for the approximations obtained from the three-variable function based on the category polynomial, the efficiency and optimality of this method. All the numerical calculations were done using a computer algebra system such as Maple. In each step, we have provided a graphical representation of the results obtained.
