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In this article, the existence of solution for the first-order nonlinear coupled system of ordinary differential equations with nonlinear coupled boundary condition (CBC for short) is studied using a coupled lower and upper solution approach. Our method for a nonlinear coupled system with nonlinear CBC is new and it unifies the treatment of many different first-order problems. Examples are included to ensure the validity of the results.

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Couette flows of an incompressible viscous fluid with non-integer order derivative without singular kernel produced by the motion of a flat plate are analyzed under the slip condition at boundaries. An analytical transform approach is used to obtain the exact expressions for velocity and shear stress. Three particular cases from the general results with and without slip at the wall are obtained. These solutions, which are organized in simple forms in terms of exponential and trigonometric functions, can be conveniently engaged to obtain known solutions from the literature. The control of the new non-integer order derivative on the velocity of the fluid moreover a comparative study with an older model, is analyzed for some flows with practical applications. The non-integer order derivative with non-singular kernel is more appropriate for handling mathematical calculations of obtained solutions.