Classical optimal single-step hybrid block techniques for ODEs: Combined basis functions with dynamic collocation strategy

Abstract

We introduce a new class of block methods based on a hybrid basis of Hermite probabilists’ polynomials and exponential polynomials. The proposed techniques exploit the complementary strengths of both families, offering enhanced accuracy, stability, and flexibility compared with schemes built on a single polynomial type. The methods employ interpolation and dynamic collocation and are formulated within a second-derivative framework. To strengthen their structure, additional terms are generated through the recurrence relation of Hermite probabilists’ polynomials, whose orthogonality provides further advantages over exponential functions. Since the accuracy of numerical methods depends largely on discretization constants, this hybridization, together with the clustered mesh points, help reduce discretization errors and error constants while maintaining stability. Rigorous theoretical analysis establishes A-stability and convergence of the schemes. Although their algebraic order of convergence is relatively low, numerical experiments demonstrate that the methods achieve improved accuracy and competitive precision factors compared with existing block approaches. These results suggest that hybrid polynomial bases provide a promising pathway for the development of robust and efficient block algorithms in numerical analysis.