Symbolic and High-Accuracy Solutions to Differential and Integral Problems via a Novel Recursive Inverse Laplace Method

In this paper, we introduce a novel semi-analytical method for solving a broad class of initial value problems involving differential, integro-differential, and delay equations, including those with fractional and variable-order derivatives. The proposed approach is based on the inverse Laplace transform, applied initially - unlike traditional Laplace-based techniques which begin with a forward transformation. By assuming the unknown solution is the Laplace transform of an auxiliary function, the method reformulates the problem in the time domain and recursively solves for this function using symbolic operations. The final solution is then obtained by applying the Laplace transform to the result. This strategy enables the construction of symbolic solutions as generalized logarithmic-power series with arbitrary accuracy, and naturally accommodates complex terms. The method is highly versatile and demonstrates superior speed and precision across a wide range of linear and nonlinear problems, including singular, fractional, and chaotic systems. Several benchmark examples are provided to validate the reliability and efficiency of the proposed technique compared to classical numerical methods. The results confirm that the new method offers a powerful and flexible framework for symbolic computation of initial value problems.