Machine Learning for Modeling the Singular Multi-Pantograph Equations

Abstract

In this study, a new approach to basis of intelligent systems and machine learning algorithms is introduced for solving singular multi-pantograph differential equations (SMDEs). For the first time, a type-2 fuzzy logic based approach is formulated to find an approximated solution. The rules of the suggested type-2 fuzzy logic system (T2-FLS) are optimized by the square root cubature Kalman filter (SCKF) such that the proposed fineness function to be minimized. Furthermore, the stability and boundedness of the estimation error is proved by novel approach on basis of Lyapunov theorem. The accuracy and robustness of the suggested algorithm is verified by several statistical examinations. It is shown that the suggested method results in an accurate solution with rapid convergence and a lower computational cost.

Application of Vieta–Lucas Series to Solve a Class of Multi-Pantograph Delay Differential Equations with Singularity

“…Finally, we consider the delayed-like terms χ 1 R pτ s ηq appeared in (1) to be written in the matrix forms. For this purpose, we first utilize (20) and then replace η by τ s η to obtain:…”

“…For the second initial condition χ 1 p0q " χ 1 , we utilize the relation (20). Similarly, we write η to 0 in (20) to have:…”

“…The main subject of this research paper was to consider a polynomial-based technique to obtain an approximate solution to the following second-order singular multi-pantograph delay differential equation (SMDDE) in the form [18][19][20]:…”

“…Recently, the generalized pantograph type differential equations have been approximately solved by using the shifted Legendre polynomial solutions, the special matrix approach, the Taylor operational matrix method, and an approach based on the combination of the variational iteration method and Laplace transform scheme [21][22][23][24]. Furthermore, singular-type differential equations with delay have been solved by the Bernoulli, Bessel, reproducing kernel and machine learning approaches, as well as by neural-swarming heuristic methods in [2,3,[18][19][20]. In addition, the collocation approach based on the Jacobi rational-Gauss collocation scheme, the stable Runge-Kutta methods, the shifted orthonormal Bernstein polynomials method and the least squares-Epsilon-Ritz algorithm were introduced for solving generalized pantograph equations [25][26][27][28].…”

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

“…where 0 < ϖ 1 , ϖ 2 ≤ 1,I 1 ,I 2 are finite constants; Λ 1 ,Λ 2 are analytical functions; and D ω τ is the Caputo fractional derivative (CFD). The FMPS is a type of DDE that arises in a number of physics and engineering applications, including quantum mechanics, dynamical systems, electronic systems, and population dynamics [65,66].…”