An Equally Spaced Numerical Method for Solving Volterra Integro-Differential Equations of Second Kind

The researchers created an Equally Spaced New Numerical Method (NNM), which functions as a highly effective tool for solving Volterra integro-differential equations of the second kind that scientists use to model physical, engineering, and biological systems with memory effects. The new method derives from using a linear block algorithm, which implements the third derivative together with a linear multistep framework to construct a hybrid block scheme that operates on equally spaced nodes. The formulation delivers two numerical schemes that operate continuously and discretely, and their coefficients are derivable through analytical methods. The method undergoes a comprehensive theoretical examination, which establishes its fundamental characteristics, including order and error constant, together with consistency, zero-stability, and convergence and region of absolute stability. The method evaluation uses numerical tests on standard problems, which demonstrate that the NNM produces results that match analytical solutions while achieving lower error rates than existing numerical methods, which use trapezoidal-type and block methods. The NNM functions as a dependable and competitive numerical method for Volterra integro-differential equations of the second kind, according to the results, which support its accuracy and stability and robustness demonstrated through tabulated data and graphical representations.