pantograph system with mixed riemann-liouville and‎ ‎caputo-hadamard‎ ‎sequential‎ ‎fractional derivatives‎: ‎existence and ulam-stability

‎the pantograph equation improves the mathematical model of the system includes modeling the motion of the wire connected with the dynamics of the supports and modeling the dynamics of the pantograph‎. ‎the subject of this paper is the existence and ulam stability of solutions for a coupled system of sequential pantograph equations of fractional order involving both riemann-liouville and caputo-hadamard fractional derivative operators‎. ‎by applying the classical theorems in nonlinear analysis‎, ‎such as the banach’s fixed point theorem and leray-schauder nonlinear alternative‎, ‎the uniqueness and existence of solutions are obtained‎. ‎furthermore‎, ‎the ulam stability results are also presented‎. ‎finally‎, ‎we have shown the results in the applications section by presenting various examples to numerical effects which provided to support the theoretical findings.