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The most significant objective of this article is the adoption of a method with a free parameter known as "The Optimum Asymptotic Homotopy Method" which has been utilized in order to obtain solutions for integral differential equations of high-order non integer derivative. The process in this method is more favorable than "Homotopy Perturbation Method" as it has a more rapid convergence compared to the mentioned method or even the similar methods. Another advantage of this method is that the convergence rate is recognized as control area. It is worth mentioning that Caputo derivative is adopted in this article. A number of instances are provided to better understand the method and its level of efficiency compared to other same methods. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the solution of the Riccati differential equations of fractional order (FRDEs) involving Caputo derivative (CD), Caputo-Fabrizio derivative (CFD) or Atangana-Baleanu derivative (ABD) using a semi-analytical iterative approach. Temimi and Ansari introduced this method and called it TAM. The linearization of the method involves splitting the problem into a linear and nonlinear part. While this approach is a semi-analytical iterative technique, there is a significant amount of analytical work where the computational times of integrals are needed to be carried out numerically. The novelty and the motivation behind this paper is the comparison of the time used in minutes which is given for three derivatives: CD, CFD and ABD. Meanwhile, the comparison of the approximate solutions with CD, CFD and ABD are presented. With the help of the software Mathematica, the computational results have been obtained. [ABSTRACT FROM AUTHOR]

Many numerical methods have been developed for nonlinear fractional integro-differential Volterra's population model (FVPG). In these methods, to approximate a function on a particular interval, only a restricted number of points have been employed. In this research, we show that it is possible to use the fuzzy transform method (F-transform) to tackle with FVPG. It makes the F-transform preferable to other methods since it can make full use of all points on this interval. We also make a comparison showing that this method is less computational and is more convenient to be utilized for coping with nonlinear integro-differential equation (IDEs), fractional nonlinear integro-differential equation (FIDEs), and fractional ordinary differential equations (FODEs). [ABSTRACT FROM AUTHOR]

In this article, we have applied Jacobi polynomial to solve Riccati differential equation of fractional order. To do so, we have presented a general formula for the Jacobi operational matrix of fractional integral operator. Using the Tau method, the solution of this problem reduces to the solution of a system of algebraic equations. The numerical results for the examples presented in this paper demonstrate the efficiency of the present method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following problems: 1. 'Let à is a fuzzy number and x a real number. Then, what do we say about whether x is greater (smaller) than or equal to �'. 2. 'Let à and $\tilde{B}$ are two fuzzy numbers. Then, what do we say about whether $\tilde{B}$ is greater (smaller) than or equal to �'. [ABSTRACT FROM AUTHOR]

Ranking generalized fuzzy numbers plays an important role in many applied models and, in particular, decision-making procedures. In ranking process of two generalized fuzzy numbers, it is natural to compare the sets of values in support of two generalised fuzzy numbers. Accordingly, the comparison of a real number and a generalised fuzzy number as well as two generalised fuzzy numbers have to be considered. On the other hand, it is seen that a definitive process of comparison of a real number and a generalised fuzzy number, as well as two generalised fuzzy numbers, is not possible. So in this study, a method for comparing a real number and a generalised fuzzy number with a degree of accuracy (between a zero and one) is defined and then the method is generalized to compare two generalised fuzzy numbers. In general, an index to rank a real number and generalised fuzzy number is constructed. Eventually, this index is extended to rank two generalised fuzzy numbers based on the concept of accuracy of comparison. The advantage of our method is that it can compare two generalised fuzzy numbers with an accuracy of comparison. Also, a definition is introduced to make a definitive comparison. Finally, the proposed method is illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]

The present work reports an attempt to show the plausibilility of applying fuzzy transform method (FTM) to work out an approximate solution for space-time differential of non integer order which is named advection-dispersion equation (STFADE). From this perspective, the derivatives of non integer order are viewed from the Caputo sense. In approximate approaches, just a certain limitted number of points are utilized to work out an approximate function for a given distance. Nontheless, the main reason why the F-transform is prefered to other similar approaches is that it takes advantage of whole the existing points in the distance. The achieved numerical results indicate that the suggested algorithm results in appropriate approximate solutions. [ABSTRACT FROM AUTHOR]

An approach for searching an approximate solution of high-order integro-differential equations featuring fractional derivatives has been proposed in this paper. Initially, using trigonometric basic functions (TBFs), we determine the transformation functions in association to TBFs. Next, the approximate function is presented as a combination of TBFs and transform functions. The convergence of this approach is also presented. Using discrete derivatives of the solution to gain an approximate solution, we find the approximate solution, which satisfies the high-order integro-differential equations featuring fractional derivatives. An algorithm of this approach is applied for various examples, and one example is illustrated in detail. [ABSTRACT FROM AUTHOR]

We have perused about the existence of a solution toward hybrid initial value problem (HIVP) featuring fractional q-derivative ... in which D948q denotes the Riemann-Liouville fractional q-derivative in the order of δ. In Banach algebra, by making use of a fixed point theorem based Dhage along with mixed Lipschitz and Caratheodory condition, a way of solving the above fractional Hybrid initial value problem (FHIVP) featuring q-derivatives verified, in this study. [ABSTRACT FROM AUTHOR]

Looking for offer, an approximate solution of the Riccati's differential equations in this research work is based on a fuzzy linguistic model (FLM). It is shown by the FLM that it is possible to approximate continuous functions which are satisfied at the interpolation points. To obtain the function as the approximation of the solution of the RDEs, fuzzy linguistic model (FLM) was used. In numerical approaches, in order to find an approximation for a function that exists on a particular interval, just a limited number of points are utilized. However, the distinctive feature of the FLM in comparison with other approaches is that in this method all points in the arbitrary interval are used. Five examples along with their approximate solutions have been presented to show the suitability, convenience, simplicity and performance of the method. The outcomes demonstrate that the advantage of this approach compared to other methods is that there are no limiting conditions for nonlinear differential equations of order one or two with initial conditions or boundary conditions. [ABSTRACT FROM AUTHOR]