numerical method

In the current study, we design a new computational method to solve a class of Li´enard’s equations. This equation originates from advancements in radio and vacuum tube technology. To attain the proposed goal, we develop a method using a three-layer artificial neural network, consisting of an input layer, a hidden layer, and an output layer. We use the Morgan–Voyce even Fibonacci polynomials and sinh function as activation functions for the hidden layer and the output layer, respectively. Then, the neural network is trained using a classical optimization method. Finally, we an-alyze four examples using graphs and tables to demonstrate the accuracy and effectiveness of the numerical approach.

In this paper, we proposed a numerical method based on the shifted fractional order Jacobi and trapezoid methods to solve a type of distributed partial differential equations. The fractional derivatives are considered in the Caputo-Prabhakar type. By shifted fractional-order Jacobi polynomials our proposed method can provide highly accurate approximate solutions by reducing the problem under study to a set of algebraic equations which is technically simpler to handle. In order to demonstrate the error estimates, several lemmas are provided. Finally, numerical results are provided to demonstrate the validity of the theoretical analysis.

We deal with some effective numerical methods for solving a class of nonlinear singular two-point boundary value Fredholm integro-differential equations. Using an appropriate interpolation and a q-order quadrature rule of integration, the original problem will be approximated by the non-linear finite difference equations and so reduced to a nonlinear algebraic system that can be simply implemented. The convergence properties of the proposed method are discussed, and it is proved that its convergence order will be of O(hmin{ 72 ,q− 12 }). Ample numerical results are addressed to con-firm the expected convergence order as well as the accuracy and efficiency of the proposed method.

In this study, we construct the fractional-order Bernstein wavelets for solving stochastic fractional integro-differential equations. Fractional-order Bernstein wavelets and their properties are presented for the first time.  The fractional integral operator of fractional-order Bernstein wavelets together with the Gaussian integration method is applied to reduce stochastic fractional integro-differential equations to the solution of algebraic equations which can be simply solved to obtain the solution of the problem.  Also, an error estimation for our approach is introduced.  The numerical results demonstrate that our scheme is simply applicable, efficient, powerful and very precise at the small number of basis functions.

In this paper, an efficient method is developed for the approximate solution of a benchmark non-smooth dynamical system. In the proposed method, the trapezoidal method is utilized for solving the Tacoma Narrows Bridge equation. For this purpose, at first, the integral form of the dynamical equation is considered. Afterwards, the obtained integral equation is discretized by the trapezoidal method. The accuracy and performance of the proposed method are examined by means of some numerical experiments.

A new method for solving n-order fuzzy differential equation by using polynomial interpolationA new method for solving n-order fuzzy differential equation by using polynomial interpolationGiven the importance of the role of fuzzy differential equations in science and engineering,in this paper, we study a numerical method for solving N th order fuzzy differential equations under generalized differentiability. In this method a solution of fuzzy differential equation is approximated by fuzzy polynomial in the form of piece wise fuzzy polynomials in eachsub interval of interval solution. In special case, for solving second order fuzzy differential equation under generalized differentiability, according to the type of differentiability, four cases are considered, then fuzzy polynomial approximation in each cases for solving fuzzy differential equation were constructed. The order of the piece wise fuzzy polynomial in each sub interval of solution is two .Finally this method is illustrated by solving two second order fuzzy differentialequations under generalized differentiability.

In this paper, we consider the Brato differential equation, in which the boundary condition values ​​are fuzzy values and the purpose is to calculate the approximate Solution. For this, first, using arithmetic operations on fuzzy data, we convert the Bratu differential equation into three sets of differential equations with exact value, and then, using the Tamimi and Ansari method (TAM), the approximate solution of the differential equation can be calculated. Finaly, two examples to express efficiency and simplicity by finding an approximate solution have been presented. Matmetica software has been used for all calculations and plots.

The Newell-Whitehead-Segel (NWS) equation is an important model arising in fluid mechanics. Various researchers worked on approximate solutions to this model by using different methods. In this paper, the Sine-Cosine wavelets method is applied for solving numerically the NWS equation. The Sine-Cosine wavelet operational matrix of integration is obtained and used to transform the equations into a system of algebraic equations. To demonstrate the effectiveness and applicability of this method, two numerical examples are included.

In recent years many numerical methods have proposed for solving linear integral equations. In this paper we propose the finite differences method for solving linear and nonlinear Fredholm integral equations of the second kind with arbitrary kernel and present some examples to illustrate this method.

Dynamical systems with delay are widespread in nature. The study of time-delay induced changes in the collective behavior of systems of coupled nonlinear oscillators is a subject of great interest, both because of its fundamental importance from the point of view of dynamical systems and because of its practical applications. In this paper, an explicit technique is proposed for numerical solution of nonlocal dynamical systems with time delay. The proposed method is adopted quadratic spline interpolation. Then, the error analysis of the developed method is discussed. It is exploited in the discussion of nonlocal delay Ikeda and Hutchinson models. Finally, the performance of the presented approach is verified by applying the error and convergence study for different values of fractional order parameters.

Forex indicators are one way of examining market data. One of the best methods is using approximations of quadratic curves. For example, the Parabolic SAR indicator is an indicator that approximates the trend by a parabola. Of course, not all indicators necessarily map the direction of the trading trend with a curve, but discovering a clear curved path for the trading trend is definitely the dream of many great analysts. The problem that we are dealing with here is actually trying to generalize the idea of approximating the direction of the trading process by using hyperbolic functions. Suppose there are two fixed points in the movement of the financial market and the path length between these two points is also a fixed number, using numerical approximation methods, we try to calculate the maximum and minimum amount of financial market movement. The main idea that is in our minds is the approximation of financial market trends by using hyperbolic functions. We approximate the path between these two points with a hyperbolic sine function. Finally, we try to find the maximum or minimum points of this hyperbolic sine function numerically.

In this paper we apply the Chebyshev polynomials method for the numerical solution of a class of variable-order fractional integro-differential equations with initial conditions. Moreover, a class of variable-order fractional integro-differential equations with fractional derivative of Caputo-Prabhakar sense is considered. The main aim of the Chebyshev polynomials method is to derive four kinds of operational matrices of Chebyshev polynomials. With such operational matrices, an equation is transformed into the products of several dependent matrices, which can also be viewed as the system of linear equations after dispersing the variables. Finally, numerical examples have been presented to demonstrate the accuracy of the proposed method, and the results have  been compared with the exact solution.

In this paper, an efficient numerical method is used to provide the approximate solution of distributed-order fractional partial differential equations (DFPDEs). The proposed method is based on the fractional integral operator of fractional-order Bernoulli-Legendre functions and the collocation scheme. In our technique, by approximating functions that appear in the DFPDEs by fractional-order Bernoulli functions in space and fractional-order Legendre functions in time using Gauss-Legendre numerical integration, the under study problem is converted to a system of algebraic equations. This system is solved by using Newton's iterative scheme, and the numerical solution of DFPDEs is obtained. Finally, some numerical experiments are included to show the accuracy, efficiency, and applicability of the proposed method.

Micromixers are an important part of microfluidics systems. In the present work, mixing was enhanced through the three helix types of micromixers. As a result of Dean vortices, a mixing index of 99% obtained at a very short length of the micromixers for the Reynolds number of 10. It is also obtained that the micromixer with rectangular cross-section showed better enhancement compared to the circular and triangular cross-section.

In the current investigation, the distributed order fractional derivative  operational matrix based on the  Legendre wavelets (LWs) as the basis functions is derived. This operational matrix is applied together with collocation method for  solving  distributed order fractional differential equations. Also, convergence analysis of the proposed scheme is given. Finally, numerical examples are presented to show the efficiency and superiority  of the mentioned scheme.

This paper presents an approximate method to solve a class of fractional partial differential equations (FPDEs). First, we introduce   radial basis functions (RBFs) combined with wavelets.  Next, we obtain fractional integral operator (FIO) of wavelets-radial basis functions (W-RBFs) directly.  In the next step, the W-RBFs and their FIO  are used to transform the problem under consideration into a  system of algebraic equations, which can be simply solved to achieve the solution of the problem.   Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the method.

In this paper, we use Chebyshev polynomials to seek the numerical solution of a class of multi-variable order fractional differential equation (MVODEs) that the fractional derivative is described in the Caputo-Prabhakar sense. Using operational matrices, the original equations are transferred to a system of algebraic equations. By solving the system of equations, the numerical solutions are acquired that this system may be solved numerically using an iterative algorithm. The effectiveness and convergence analysis of the numerical scheme is illustrated through four numerical examples.

In this paper a numerical method for solving forth order fuzzy di erential equations under generalized differentiability is proposed. This method is based on the interpolating a solution by piecewise polynomial of degree 8 in the range of solution . We investigate the existence and uniqueness of solutions. Finally a numerical example is presented to illustrate the accuracy of the new technique.