We consider a class of nonlinear integro-differential equations involving a fractional power of the Laplacian and a nonlocal quadratic nonlinearity represented by a singular integral operator. Initially, we introduce cut-off versions of this equation, replacing the singular operator by its Lipschitz continuous regularizations. In both cases we show the local existence and global uniqueness in L...

Abstract This research apparatuses an approximate spectral method for the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel (TFPIDE). The main idea of this approach is to set up new Hilbert space that satisfies initial and boundary conditions. collocation applied obtain precise numerical approximation using basis functions based on shifted first-kind ...

Abstract The shifted Chebyshev polynomials of the fifth kind (SCPFK) and collocation method are employed to achieve approximate solutions a category functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for numerical solution problem under study. suggested changes s...

We investigate solving partial integro-differential equations (PIDEs) using unsupervised deep learning. The PIDE is employed for option pricing, when the underlying process a Lévy with jumps. learning approach employs neural network as candidate solution and trains to satisfy PIDE. By matching boundary conditions, would yield an accurate Unlike supervised learning, this does not require any lab...

This paper contributes to investigate the Jacobi spectral and pseudo-spectral Galerkin techniques solve a general form of nonlinear weakly singular Volterra integro-differential equations first order. By applying some suitable change variables, we have made solution mentioned be smooth. Then, by schemes, accurate solutions are computed efficiently. Rigorous convergence analysis associated with ...

A high-order finite difference numerical scheme based on the compact operator is proposed in this paper for time-fractional partial integro-differential equations with a weakly singular kernel, where derivative term defined Riemann-Liouville sense. Here, stability and convergence of constructed are proved L∞ norm, accuracy order O(τ2+h4), τ h temporal spatial step sizes, respectively. The advan...

We construct the algebra of integro-differential operators over an ordinary integro-differential algebra directly in terms of normal forms. In the case of polynomial coefficients, we use skew polynomials for defining the integro-differential Weyl algebra as a natural extension of the classical Weyl algebra in one variable. Its normal forms, algebraic properties and its relation to the localizat...