Solving singularly perturbed differential-difference equations arising in science and engineering with Fibonacci polynomials

In this paper, we introduce a method to solve singularly perturbed differential-difference equations of mixed type, i.e., containing both terms having a negative shift and terms having a positive shift in terms of Fibonacci polynomials. Similar boundary value problems are associated with expected first exit time problems of the membrane potential in the models for the neuron. First, we present some preliminaries about polynomial interpolation and properties of Fibonacci polynomials then a new approach implementing a collocation method in combination with matrices of Fibonacci polynomials is introduced to approximate the solution of these equations with variable coefficients under the boundary conditions. Numerical results with comparisons are given to confirm the reliability of the proposed method for solving these equations.