Authors:
Asmaa A. Mahdi,Ruqaia Jwad Kadhim,Hasanain Jalil Neamah Alsaedi,Adel S. Hussain,Rana Aziz Yousif Almuttalibi,DOI NO:
https://doi.org/10.26782/jmcms.2026.09.00005Keywords:
Stochastic Integro-differential equations,Laplace Adomian decomposition method,Laplace transform,Adomian polynomial,Physics-Informed Neural Networks (PINNs).,Abstract
We generalize the Laplace Adomian Decomposition Method (LADM) to solve both linear and nonlinear stochastic integro-differential equations that are constructed on arbitrary time scales. The presented method implements the time-scale Laplace transform to transform delta-derivative terms into the Laplace domain and build an Adomian series of the nonlinearities, and uses the inverse transform to obtain time-domain solutions; the stochastic integrals not in closed form are approximated by Physics-Informed Neural Networks (PINNs) trained to satisfy Ito-type integral equations. On illustrative linear and nonlinear Volterra problems, we validate the method where the LADM series recovers solutions to the ADM equations within the truncation error (examples show agreement to ?10?^(-3)in mean absolute error) with fewer explicit evaluations of integrals than ADM (reduction in the number of convolution/evaluation steps, approximately 4070 of examples). At points where stochastic integrals are approximated, PINNs give path-wise means that are similar to Monte-Carlo estimates (Table 3). We make the following contributions: (i) we provide a time-scale compatible LADM formulation of Ito-type stochastic forcing; (ii) we show computational savings of LADM over classical ADM on nonstandard time scales; and (iii) we provide a viable LADM-PINN hybrid that computes non-closed-form stochastic integrals. Rigorous convergence and variance-reduction constraints and guidelines are mentioned.Refference:
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