A delannoy polynomial-based collocation framework for solving nonlinear fourth-order Integro-differential equations

Abstract This paper presents a new approach using shifted Delannoy polynomials to tackle the nonlinear fourth-order integro-differential equations smoothly. We are using what makes these polynomials special to build our system, which turns the initial problem into a set of nonlinear algebraic equations we can solve with Newton’s method. We have carefully solidified the mathematics to ensure our method works, and it turns out the error gets smaller fast when dealing with smooth functions. Tests show our method is super accurate, reaching errors as small as $$10^{-16}$$ , better than other existing methods. It’s easy to set up, converges quickly, and is numerically stable. Aside from the theory, this method can be used in beam mechanics, viscoelastic systems, and other memory effect models.