A new collocation method based on boubaker polynomials and integral to solve some differential equations and pantograph equations

This study introduces a novel collocation approach utilizing Boubaker polynomials and integration techniques to address differential equations and pantograph-type problems. Initially, the dominant derivative term in the equation is represented through a truncated Boubaker polynomial series, and this representation is cast into matrix form. Subsequently, successive integrations of the initial approximation are carried out to derive the matrix expressions corresponding to the remaining derivatives and the unknown function involved in the problem. After approximating the highest-order derivative, the conditions specified in the differential equation are incorporated at each stage of integration. Once the matrix expressions of the unknown function and its derivatives are inserted into the original equation, the problem is transformed into a system of algebraic equations by employing uniformly distributed collocation points. When the original differential problem is linear, the resulting algebraic system also remains linear; in contrast, a nonlinear differential problem yields a nonlinear algebraic system. Solving this system provides the values of the unknown coefficients used in the approximation. These coefficients simultaneously define the solution structure of the unknown function, and their substitution into the approximate expression yields the final solution. This new method is applied, in this thesis, to higher-order linear differential equations, first-order pantograph equations, multi-pantograph equations, the Lane–Emden differential equation, the Riccati differential equation, generalized pantograph equations, and the logistic population model problems. Additionally, after constructing a residual function and formulating an error problem, this error problem is also solved using the same method to estimate the error. To evaluate the method and its error estimation framework, it is applied to selected test problems. The resulting solutions are benchmarked against well-known approaches from the literature to assess its effectiveness.