Progress in Falkner-Type Method for Second Order Initial Value Problems in ODEs

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This study centers on the development of block hybrid techniques utilizing power series as fundamental functions, achieved through a combination of interpolation and collocation methodologies. The primary objective is to provide a numerical solution for second-order initial value problems within the realm of ordinary differential equations. The formulated block hybrid method is characterized by a step count of k=2, featuring two off-step points and four additional off-step points.

A comprehensive analysis was conducted to assess the fundamental attributes of these numerical methods. The investigation revealed several key properties, including consistency, zero-stability, and convergence. These favorable qualities establish the suitability of the methods for effectively addressing a diverse array of problems. Notably, the methods exhibit proficiency in solving various problem classes, encompassing linear and non-linear scenarios, oscillatory phenomena, dynamic complexities, and stiff systems.

The empirical outcomes derived from the proposed techniques demonstrate their heightened accuracy and comparative advantages when contrasted with existing methods discussed in the existing literature. Through careful evaluation, it becomes evident that the newly introduced methods offer enhanced precision and efficacy in solving the target problem types. Overall, this research serves to advance the domain of numerical problem-solving by introducing innovative block hybrid techniques founded on power series, thereby presenting a robust solution strategy for a broad spectrum of second-order initial value problems in ordinary differential equations.

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