In this paper, we introduce second derivative multistep collocation meth-ods for the numerical integration of ordinary differential equations (ODEs). These methods combine the concepts of both multistep methods and col-location methods, using second derivative of the solution in the collocation points, to achieve an accurate and efficient solution with strong stability properties, that is, A-stability for ODEs. Using the second-order deriva-tives leads to High order of convergency in the proposed methods. These methods approximate the ODE solution by using the numerical solution in some points in the r previous steps and by matching the function values and its derivatives at a set of collocation methods. Also, these methods utilize information from the second derivative of the solution in the colloca-tion methods. We present the construction of the technique and discuss the analysis of the order of accuracy and linear stability properties. Finally, some numerical results are provided to confirm the theoretical expecta-tions. A stiff system of ODEs, the Robertson chemical kinetics problem, and the two-body Pleiades problem are the case studies for comparing the efficiency of the proposed methods with existing methods.

The main objective of this work is to present a High-order numerical method to solve a class of nonlinear Fredholm integro-differential equations. By multiplying appropriate efficient factors and constructing an appropriate approximate function,  as well as employing a numerical integration method of order $\gamma$, the above-mentioned problem can be simplified to a nonlinear system of algebraic equations. Furthermore, we discuss the convergence analysis of the presented method in detail and demonstrate that it converges with an order $\mathcal{O}(h^{3.5})$ in the $L^2$-norm. Some test examples are provided to demonstrate that the claimed order of convergence is obtained.

Introduction: The modeling of many real-life physical systems leads to a set of fractional differential equations. Also fractional differential equations appear in various physical processes such as viscoelasticity and viscoplasticity, modeling of polymers and proteins, transmission of ultrasound waves, signal processing, control theory, etc. Most of fractional differential equations especially their nonlinear types do not have exact analytic solution, so numerical methods must be used. Therefore many authors have worked on the numerical solutions of this kind of equations. In recent years, many numerical methods have emerged, such as, the Adomian decomposition method, the Homotopy method, the multistep method, the extrapolation method, the spline collocation method, the product integration method and the predictor-corrector method. But most of the aforementioned methods consider the linear type of equations without a reliable theoretical justification. Then providing an efficient numerical scheme to approximate the solutions of nonlinear fractional differential equations is worthwhile and new in the literature. The main object of this paper is to develop and analyze a High order numerical method based on the collocation method when applies the orthogonal Jacobi polynomials as bases functions for the single term nonlinear fractional differential equations. Material and methods: Due to the well-known existence and uniqueness theorems the solutions of the fractional differential equations typically suffer from singularity at the origin. Consequently direct application of the Jacobi collocation method may lead to very weak numerical results. To fix this difficulty, we introduce a smoothing transformation that removes the singularity of the exact solution and enables us to approximate the solution with a satisfactory accurate result. Convergence analysis of the proposed scheme is also presented which demonstrates that the regularization process improves the smoothness of the input data and thereby increases the order of convergence. Results and discussion: We illustrate some test problems to show the effectiveness of the proposed scheme and to confirm the obtained theoretical predictions. In overall, the reported results justify that the proposed regularization strategy works well and the obtained approximate solutions have a good accuracy. To show the applicability of our approach we solve a practical example which is developed for a micro-electrical system (MEMS) instrument that has been designed primary to measure the viscosity of fluids that are encounter during oil well exploration using the proposed scheme. Moreover, we make a comparison between our scheme and the operational Tau method to show the efficiency of our technique. The reported results approve the superiority of the proposed approach. Finally, we consider a problem that we do not have access to its exact solution. In this case, we use the “ Variational Iteration Method (VIM)” as a qualitatively correct picture of the exact solution (the source solution) to evaluate the precision of the proposed technique. The obtained results approve that our approach produces the approximate solution which is in a good agreement with source ones. Conclusion: The following conclusions were drawn from this research. A reliable numerical method based on the Jacobi collocation method to approximate the solutions of a class of nonlinear fractional differential equations was developed. To achieve an efficient approximation a regularization strategy was proposed that improves the smoothness of the input data and enables us to obtain an approximate solution with a satisfactory accuracy. Convergence analysis of the proposed method was investigated which confirmed the High order of convergence of the proposed method.

The most significant objective of this article is the adoption of a method with a free parameter known as \The Optimum Asymptotic Homotopy Method" which has been utilized in order to obtain solutions for integral differential equations of High-order non integer derivative. The process in this method is more favorable than \Homotopy Pertur-bation Method" as it has a more rapid convergence compared to the mentioned method or even the similar methods. Another advantage of this method is that the convergence rate is recognized as control area. It is worth mentioning that Caputo derivative is adopted in this article. A number of instances are provided to better understand the method and its level of efficiency compared to other same methods.

In this research, in order to investigate the effect of the piezoelectric patch which is used as a sensor or actuator in rotating flexible structures such as a helicopter blade, the free vibrations of the rotating rectangular sheet with and without the piezoelectric patch have been presented. First-order shear deformation theory is considered for plate displacement and piezoelectric field. Considering the effect of Coriolis acceleration, centrifugal acceleration and centrifugal in-plane forces, the equations of motion are derived from Hamilton's principle and the electromechanical couple equation is obtained from Maxwell's equation. For piezoelectric, two electrical conditions, open circuit and closed circuit, which are used in sensors and actuators, respectively, have been considered. The equations are discretized with the help of the numerical method of generalized differential squares and the matrices of inertia mass, eccentricity, Coriolis and stiffness matrix are obtained. Natural frequency values for beam and rotating plate have been compared in Abaqus software. Also, the values obtained from the numerical solution in MATLAB have been verified with articles and ABAQUS, which have High accuracy. The effect of parameters such as hub radius, rotation speed, sheet thickness, aspect ratio, piezoelectric patch thickness and applied voltage on the natural frequency of the system has also been investigated.