Five-Point Implicit Linear Multistep Method for Numerical Computation of Higher Order Differential Equation

Higher-order ordinary differential equations arise in a range of mathematical models in science and engineering. This study develops a five-point implicit linear multistep method for the direct numerical approximation of fourth-order initial value problems. The formulation employs a polynomial power-series approximation, together with interpolation at selected grid points and collocation of the fourth derivative. These conditions yield a continuous approximation from which a discrete five-point implicit scheme is obtained. The method is formulated without first converting the fourth-order equation into a system of first-order equations. A Taylor-series expansion of the associated difference operator is used to examine the order conditions and local truncation error. The analysis presented in the manuscript identifies a seventh-order method with an error constant of 1/3024. To illustrate numerical performance, the scheme is applied to two fourth-order test problems for which analytical solutions are stated. Approximate solution values and absolute errors are tabulated using a step size of 0.1 over the interval from 0 to 1, and graphical comparisons are provided. For the first example, the reported absolute error at the final tabulated mesh point is 1.0880*10-6. The calculations illustrate the application of the proposed discretisation to the selected test problems. Further numerical verification and assessment under additional problem settings are necessary before its broader accuracy and efficiency can be established.