Exact smooth and nonsmooth solutions for integro-partial differential equations by rapidly convergent approximation method

We investigate a general class of second-order integro–ordinary-differential equations with arbitrary-power nonlinear terms, which can be used as a mathematical model for a variety of important physical areas in mathematics, mathematical physics, and applied sciences. The exact smooth and nonsmooth solutions of the aforementioned integro–differential equation in terms of the Gauss hypergeometric function are obtained here for the first time using the rapidly convergent approximation method. The prerequisites for the existence of such solutions are outlined in a theorem. Additionally, a few theorems are presented that contain the conditions under which our derived nonsmooth solution can be viewed as a weak solution. Using the aforementioned results, we obtain exact smooth and nonsmooth solutions of the following nonlinear integro-partial differential equations: the \((1+1)\)-dimensional integro–differential Ito equation, the \((3+1)\)-dimensional Yu–Toda–Sasa–Fukuyama equation, and the Calogero–Bogoyavlenskii–Schiff equation.