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This manuscript presents a one-step method incorporating a second-derivative applied to obtain approximate solutions of first-order initial-value problems of ordinary and time-dependent partial differential equations. The new scheme is derived through interpolation and collocation approaches, and the characteristics of the obtained method are analyzed. An embedding-like technique is considered and executed in adaptive mode to get better performance of the proposed method. The new scheme is used for solving some real-life application problems to determine its efficiency and accuracy. The numerical solutions obtained show that the proposed error estimate strategy is suitable in a variable step-size mode, robust and competitive compared with the results provided by other methods in the available literature.

An adaptive one-point second-derivative Lobatto-type hybrid method for solving efficiently differential systems