Development and Implementation of a Numerical method to Solve Third-Order Initial Value Problems Using an Optimized Hybrid Volterra Integral Equation of the Second Kind

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Abstract This paper introduced the formulation and execution of a numerical technique for addressing third-order initial value issues through an optimized hybrid Volterra integral equation of the second sort. Power series and exponential fitting serve as basis functions for the development of a novel two-step optimized hybrid numerical approach, suitable for addressing stiff initial value problems in third-order ordinary differential equations. The novel method exhibits enhanced convergence characteristics and has demonstrated its effectiveness on benchmark issues. The numerical implementation exhibits diminished computing expense, increased precision, and superior stability characteristics relative to conventional approaches. The optimized hybrid block designs have superior stability qualities. Numerical examples are provided to demonstrate the dependability and precision of the approximations

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