Stability and numerical solutions of higher-order nonlinear time-dependent delay differential equations using Haar wavelet collocation method

Abstract In this paper, the authors present qualitative results for the solutions of nonlinear higher-order time-dependent delay differential equations. Proof of existence and uniqueness theorem for nth order time-dependent delay differential equations using the contraction mapping theorem is presented. Stability is verified through Hyers-Ulam and Hyers-Ulam-Rassias stability theorems, using the Picard operator with the Chebyshev norm and Gronwall’s inequality. The Haar wavelet collocation method is used to find numerical solutions of higher-order time dependent delay differential equations. Two numerical examples of higher-order nonlinear delay differential equations with time-dependent delays are discussed to show efficiency and reliability of the method. Numerical results are benchmarked against existing exact solutions to validate precision and accuracy. It is observed that inaccuracies decrease with higher resolution level. Error analysis including maximum absolute error, relative error, and root mean square error, are calculated to demonstrate the robustness of the method. Further, convergence rate is calculated for each example. A detailed investigation of computational complexity, including both time and space complexity is also discussed.