Statistics

Mathematical proof

Fractional Brownian motion

Legendre polynomials

Brownian motion

Machine learning

Physics

Randomness

Banach space

Mathematical analysis

Fractional calculus

Nonlinear system

Economics

Spectral method

Algebraic equation

Geometry

Convergence (economics)

Applied mathematics

Mathematics

Computer science

Economic growth

Quantum mechanics

Scheme (mathematics)

Collocation (remote sensing)

International Journal of Modern Physics C

This paper investigates the well-posedness of a class of FSIDEs utilizing the fractional Caputo–Fabrizio derivative. Herein, the well-posedness proofs are constructed by considering some applicable conditions and combining theories of Banach space, AAT, and FPST. Approximating the solutions of such equations is still challenging for many mathematicians today due to their randomness and the hardness of finding the exact one. For the numerical aim, we introduce some useful properties of the Legendre-shifted polynomials and employ them as a basis of the collocation spectral method. The idea of this scheme is to convert such stochastic equations into algebraic systems subject to [Formula: see text]-measurable independent parameters. The stochastic term is driven by one-dimensional standard Brownian motion which is the most familiar type and for simulating its trajectories we discuss an easy method. We rigorously analyze the convergence of the proposed technique and other error behavior-bound results. Finally, various tangible numerical applications are performed to verify the present scheme’s accuracy and great feasibility and support theoretical results. The acquired results reveal that the methodology used is effective and appropriate to deal with various issues in light of the fractional Caputo–Fabrizio derivative.

Well-posedness and numerical simulations employing Legendre-shifted spectral approach for Caputo–Fabrizio fractional stochastic integrodifferential equations