Solving Nonlinear Volterra Integro-Differential Equations in the Mechanical Model of Inelastic Cables Under the Influence of Loads

In this work, two approximation techniques are presented for solving nonlinear Volterra integro-differential equations with boundary conditions, arising in the modeling of inelastic cables subjected to external loads. The foundation of the required numerical computation is established through the operational matrices of interpolating basis functions and the Gegenbauer wavelet technique, which transform the original problem into a system of algebraic equations. To construct the interpolation basis functions, orthonormal Lagrangian basis functions are employed. Subsequently, the resulting algebraic system is solved using Newton–Cotes nodes to obtain the desired numerical solution. The use of operational matrices simplifies the problem and significantly reduces the computational complexity of solving integro-differential equations. Moreover, error bounds are established, and a comprehensive convergence analysis of the proposed methods is carried out. Finally, numerical experiments supported by graphical illustrations clearly demonstrate the reliability and computational efficiency of the developed techniques.