H. R. Khodabandelo - Department of Mathematics, Imam Khomeini International University, Qazvin, Iran.
E. Shivanian - Department of Mathematics, Imam Khomeini International University, Qazvin, Iran.
S. Abbasbandy - Department of Mathematics, Imam Khomeini International University, Qazvin, Iran.

This work presents the generalized nonlinear multi-terms fractional variable-order differential 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, efficiency, accuracy of presented scheme and the flexibility 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 efficiency 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.

Nonlinear multi-terms differential equations, Fractional variable-order with proportional delays, Shifted Jacobi operational matrix, Caputo differential ‎operator.‎