A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs

This paper suggested a block-based linear multistep formula for directly solving nonlinear fourth-order initial value problems of ordinary differential equations (ODEs). The method was achieved by applying collocation and interpolation techniques to a first-kind Chebyshev polynomial. A continuous scheme was constructed through this procedure from where the proposed discrete formula was extracted. The extracted discrete formula was then implemented in block mode using the block matrix formulation and written explicitly as block equations. The proposed method is zero-stable, consistent, convergent, and p-stable, as demonstrated by the analysis of the basic properties of the derived scheme, with theoretical order eight. Six numerical examples were solved with the derived method to test its accuracy and effectiveness, all showing minimal error. A comparison with existing methods in the cited literature revealed that the proposed method offers good performance with minor errors.