The current investigations present the numerical solutions of the novel singular nonlinear fifth-order (SNFO) system of multi-pantograph differential model (SMPDM), i.e., SNFO–SMPDM. The novel SNFO–SMPDM is obtained using the sense of the second kind of typical Emden–Fowler and prediction differential models. The features of shape factor, pantograph along with singular points are provided for all four obtained classes of the SNFO–SMPDM. The extensive use of the singular models is observed in the engineering and mathematical systems, e.g., inverse systems and viscoelasticity or creep systems. For the correctness of the proposed novel SNFO–SMPDM, one case of each class is numerically handled by applying supervised neural networks (SNNs) along with the optimization of Levenberg–Marquardt backpropagation scheme (LMBS), i.e., SNNs–LMBS. A dataset using the traditional variational iteration scheme is designed to compare the proposed results of each case of SNFO–SMPDM. The obtained approximate solutions of each class using the novel SNFO-SMPDM are presented based on the training (80%), authentication (10%) and testing (10%) measures to evaluate the mean square error. Fifteen numbers of neurons, and sigmoid activation function are used in this SNN process. To authenticate the competence, and precision of SNFO–SMPDM, the numerical simulations are accessible by applying the relative measures of regression, error histogram plots, and correlation.

In this paper, a new formula of fuzzy Caputo fractional-order derivatives (0 <; v ≤ 1) in terms of shifted Chebyshev polynomials is derived. The proposed approach introduces a shifted Chebyshev operational matrix in combination with a shifted Chebyshev tau technique for the numerical solution of linear fuzzy fractional-order differential equations. The main advantage of the proposed approach is that it simplifies the problem alike in solving a system of fuzzy algebraic linear equations. An approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors is derived. Furthermore, we also discuss the convergence of the proposed method from the fuzzy perspective. Experimentally, we show the strength of the proposed method in solving a variety of fractional differential equation models under uncertainty encountered in engineering and physical phenomena (i.e., viscoelasticity, oscillations, and resistor-capacitor (RC) circuits). Comparisons are also made with solutions obtained by the Laguerre polynomials and the fractional Euler method.

In this paper, we propose a novel approach for the numerical solution of fuzzy fractional differential equations (FFDEs) under fuzzy Caputo-type derivative. More specifically, we first obtain the equivalent integral form of original problem, then the fractional integral equation is approximated using Laplace transforms. Afterwards, we can get the solution by employing any numerical method. Indeed, the proposed approach introduces an efficient and practical way to solve a wide range of fractional models under uncertainty. The most important advantage of this procedure is that the complexity of dealing with the fractional derivative is removed from the calculations, which can reduce the computational costs, considerably. Illustrative examples address the validity and appropriateness of this technique.

The main study of this paper is focused on the solutions of a class of fuzzy sequential fractional differential equations in the form of (0Dxβy)'(x) = b(x)y(x), where (0Dxβy)(x) is the fuzzy Riemann-Liouville derivative of order β ∈ (0, 1). On this subject, a new fuzzy complete metric space is introduced. Finally, we proof the existence and uniqueness of our solution using the contraction principle.

Fractional calculus has a long successful history of 300 years, as it able to model natural phenomena states more accurately than the differential equations of integer order. With this, it plays an important role in variant disciplines. Recently, variant fractional models for the Bloch equations have been proposed, however, effective numerical methods for the fractional Bloch equation (FBE) are still in the infancy stage. In this paper, we extend the time-fractional Bloch equation (TFBE) to fuzzy field under the generalized Caputo differentiability, such that these extensions have natural relationship between crisp. For this purpose, we adopted the fractional Adams-Bashforth-Moulton (FABM) type predictorcorrector method, and introduced a new variant - the fuzzy fractional ADM (FFABM) to find the numerical solution. In this case, a new theorem concerning the error of our proposed FFADM method is also presented. Finally, the capability of the newly developed numerical methods is demonstrated in a fuzzy fractional-order problem, and it achieves satisfactorily in terms of numerical stability.