in this paper, DIFFERENTIAL EQUATION of sensitivity MATRIX of required velocity with respect to position vector is derived for a linearized problem of a spacecraft motion with final velocity constraint. This MATRIX can be utilized for an implicit guidance of a spacecraft with final position and velocity constraint. The solution is obtained in the presence of aerodynamical force. Therefore, the resulting EQUATION can be used for trajectory optimization. Moreover, a change of variable is presented to reduce onboard computational burden, in addition, the method is more appropriate for analytical approaches than conventional implicit EQUATIONs.

In this paper, we propose a method to approximate the solution of a linear Fredholm integro-DIFFERENTIAL EQUATION by using the Chebyshev wavelet of the first kind as basis. For this purpose, we introduce the first Chebyshev operational MATRIX of integration. Chebyshev wavelet approximating method is then utilized to reduce the integro-DIFFERENTIAL EQUATION to a system of algebraic EQUATIONs. Illustrative examples are included to demonstrate the advantages and applicability of the technique.

This work presents the generalized nonlinear multi-terms fractional variable-order di, erential EQUATION with proportional delays. In this paper, a novel shifted Jacobi operational MATRIX technique is introduced to solve a class of these EQUATIONs mentioned, so that the main problem becomes a system of algebraic EQUATIONs that we can solve numerically. The suggested technique is successfully developed for the aforementioned problem. Comprehensive numerical tests are provided to demonstrate the generality, e, ciency, accuracy of presented scheme and the exibility of this technique. The numerical experiments compared it with other existing methods such as Reproducing Kernel Hilbert Space method (RKHSM). Comparing the results of these methods as well as comparing the current method (NSJOM) with the true solution, indicating the validity and e, ciency of this scheme. Note that the procedure is easy to implement and this technique will be considered as a generalization of many numerical schemes. Furthermore, the error and its bound are estimated.