Most of fractional DIFFERENTIAL EQUATIONs are considered on a fixed INTERVAL. In this paper, we consider a typical fractional DIFFERENTIAL EQUATION on a symmetric INTERVAL $[-\alpha,\alpha]$, where $\alpha$ is the order of fractional derivative. For a positive real number α we prove that the solutions are  $T_{n,\alpha}(x)=(\alpha+x)^\frac{1}{2}Q_{n,\alpha}(x)$ where $Q_{n,\alpha}(x)$ produce a family of orthogonal polynomials with respect to the weight function$w_\alpha(x)=(\frac{\alpha+x}{\alpha-x})^{\frac{1}{2}}$ on $[-\alpha,\alpha]$. For integer case $\alpha = 1 $, we show that these polynomials coincide with classical Chebyshev polynomials of the third kind. Orthogonal properties of the solutions lead to practical results in determining solutions of some fractional DIFFERENTIAL EQUATIONs.

Generally, in most applications of engineering, the parameters of the mathematical models are considered deterministic. Although, in practice, there are always some uncertainties in the model parameters,these uncertainties may be made wrong representation of the mathematical model of the system. These uncertainties can be generated from different reasons like measurement error, inhomogeneity of the process, chaotic behavior of systems, etc. This problem leads researchers to study these uncertainties and propose solutions for this problem. The iterative analysis is a method that can be utilized to solve these kinds of problems. In this paper, a new combined method based on INTERVAL chaotic and iterative decomposition method is proposed. The validation of the proposed method is performed on a chaotic Rossler system in stable INTERVALs. The simulation results are applied on 2 practical case studies and the results are compared with the INTERVAL Chebyshev method and RungeKutta method of order four (RK4) method. The final results showed that the proposed method has a good performance in finding the confidence INTERVAL for the Rossler models with INTERVAL uncertainties,the results also showed that the proposed method can handle the wrapping effect in a better manner to sharpen the range of non-monotonic INTERVAL.

In this paper, an extension of DIFFERENTIAL Transformation Method (DTM) which is an analytical-numerical method for solving the fuzzy partial DIFFERENTIAL EQUATION (FPDE) by using the strongly generalized differentiability concept is investigated. The proposed algorithm is illustrated by numerical example.

In this paper, we intend to solve special kind of ordinary DIFFERENTIAL EQUATIONs which is called Heun EQUATIONs, by converting to a corresponding stochastic DIFFERENTIAL EQUATION (S.D.E.). So, we construct a stochastic linear EQUATION system from this EQUATION which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding DIFFERENTIAL EQUATIONs.

To solve fuzzy DIFFERENTIAL EQUATIONs driven by Liu process, three Milstein schemes are proposed in this work, which are explicit Milstein scheme, semi-implicit Milstein scheme and improved Milstein scheme. Improved Milstein scheme is constructed by correcting the error with semi-implicit method, the error is the difference between the exact solution of fuzzy DIFFERENTIAL EQUATIONs and the solution derived from Milstein scheme. These numerical methods are proved to have strong convergence with order two. Accompanying the results above, the concept of mean-stability of numerical schemes for fuzzy DIFFERENTIAL EQUATIONs is put forward and analyzed. For a linear test EQUATION, it is showed the mean-stability region of improved Milstein scheme is bigger than explicit Milstein scheme. Finally, the accuracy and effectiveness of these schemes are confirmed through numerical examples.