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The main aim of this study is to apply the Gegenbauer polynomials for the solution of high-order linear differential difference equations with functional arguments under initial-boundary conditions.The technique we have used is essentially based on the truncated Gegenbauer series and its matrix representations along with collocation points. Also, by using the Mean-Value Theorem and residual function, an efficient error estimation technique is proposed and some illustrative examples are presented to demonstrate the validity and applicability of the method.

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A Numerical Approach with Residual Error Estimation for Eolution of High-order Linear Differential-difference Equations by Using Gegenbauer Polynomials