An effective method for solving nonlinear fractional differential equations

A new technique based on beta functions is applied to compute the exact formula for the Riemann–Liouville fractional integral of the fractional-order generalized Chelyshkov wavelets. An approximation method based on the wavelets is proposed to effectively solve nonlinear fractional differential equations. Illustrative examples show that the proposed method gives solutions with less errors in comparison with the previous methods.

     

A fractional variational iteration method for solving fractional nonlinear differential equations

Recently, fractional differential equations have been investigated by employing the famous variational iteration method. However, all the previous works avoid the fractional order term and only handle it as a restricted variation. A fractional variational iteration method was first proposed in [G.C. Wu, E.W.M. Lee, Fractional variational iteration method and its application, Phys. Lett. A 374 (2010) 2506–2509] and gave a generalized Lagrange multiplier. In this paper, two fractional differential equations are approximately solved with the fractional variational iteration method.     

A multiprocessor architecture for solving nonlinear partial differential equations

This paper presents the architecture of a special-purpose multiprocessor system, which we call the Broadcast Cube System (BCS), for solving non-linear Partial Differential Equations (PDEs). The BCS has the following important features: (a) Being based on the conceptually simple bus interconnection scheme it is easily understood. The use of homogeneous Processing Elements (PEs) which can be realized as standard VLSI chips makes the hardware less costly. (b) The interconnection network is simple and regular. The network can easily be extended to vast number of PEs by adding buses with new PEs on them and by slightly increasing the number of PEs on existing buses. The interconnection pattern is highly redundant to support fault tolerance in the event of PE failures. (c) In terms of the switching delays, the delay a message undergoes between a pair of PEs connected to a common bus is zero. The maximum delay between any pair of PEs is one unit and thus a strong localization of communicating tasks is not needed to avoid long message delays even in networks of thousands of PEs. The effectiveness of the BCS has been demonstrated by both analytical and simulation methods using heat transfer and fluid flow simulation, which requires solution of non-linear PDEs, as a benchmark program.     

多模态多目标差分进化算法求解非线性方程组

针对当前算法在求解非线性方程组时面临解的个数不完整、精确度不高、收敛速度慢等问题进行了研究,提出一种多模态多目标差分进化算法。首先将非线性方程组转换为多模态多目标优化问题,初始化一个随机种群并对种群中全部个体进行评价;然后通过非支配解排序和决策空间拥挤距离选择机制,挑选种群中的一半优质个体进行变异;接着在变异过程中采用一种新的变异策略和边界处理方法以增加解的多样性;最后通过交叉和选择机制使优质个体进行进化,直到搜索到全部最优解。在所选测试函数集和工程实例上的实验结果表明,该算法能有效地搜索到非线性方程组的解,并通过与当前四个算法进行比较,该算法在解的数量和成功率上具有优越性。     

Global stability for separable nonlinear delay differential equations

Consider the following separable nonlinear delay differential equation

, where we assume that, there is a strictly monotone increasing function f(x) on (−∞, +∞) such that

In this paper, to the above separable nonlinear delay differential equation, we establish conditions of global asymptotic stability for the zero solution. In particular, for a special wide class of f(x) which contains a case of f(x) = e^x−1, we give more explicit conditions. Applying these, we offer conditions of global asymptotic stability for solutions of nonautonomous logistic equations with delays.     

Application of a differential transformation method to strongly nonlinear damped q-difference equations

The theory of approximate solution lacks research on the area of nonlinear $q$-difference equations. This article explores the possibility of using the differential transformation method to find an approximate solution for strongly nonlinear damped $q$-difference equations. The time response of the nonlinear equation is presented under different parameter conditions, and the results are then compared with those derived from the numerical method to verify the accuracy of the proposed method.     

Using the iterative reproducing kernel method for solving a class of nonlinear fractional differential equations

In previous works, we have devoted to using the reproducing kernel methods solving integer order differential equations, based on the review of previous works, in this paper, we mainly present a method for solving a class of higher order fractional differential equations with general boundary value problems by using Taylor formula into reproducing kernel space. Its analytical solution is represented in the form of series. The analytical solution and approximate solution obtained by this method is given and it is uniformly converge to the exact solution and its corresponding derivatives. The numerical examples are studied to demonstrate the accuracy of the present method.     

A numerical method for solving systems of nonlinear equations

An iterative solution method for systems of nonlinear equations is proposed, making use of a nonlinear technique for the construction of the approximations. The method is efficient for solving quadratic equations.Translated from Kibernetika, No. 3, pp. 60–64, 69, May–June 1990.     

求解非线性方程重根的区间牛顿法

讨论了求解非线性方程重根问题,针对此时Moore区间牛顿法不再适用,以及Hansen改进的区间牛顿法收敛速度慢的情况,通过引入原方程的一种相关方程,建立了求解非线性方程重根的区间牛顿法;证明了其局部平方收敛的性质,给出了数值算例。验证了新算法比Hansen改进的区间牛顿法具有更快的收敛速度,且算法是有效和可靠的。     

A new method for solving fuzzy linear differential equations

In this paper, a novel operator method is proposed for solving fuzzy linear differential equations under the assumption of strongly generalized differentiability. To this end, the equivalent integral form of the original problem is obtained then by using its lower and upper functions the solutions in the parametric forms are determined. The proposed method is illustrated with numerical examples.     

Global bifurcations of strongly nonlinear oscillator induced by parametric and external excitation

The global bifurcation of strongly nonlinear oscillator induced by parametric and external excitation is researched. It is known that the parametric and external excitation may induce additional saddle states, and result in chaos in the phase space, which cannot be detected by applying the Melnikov method directly. A feasible solution for this problem is the combination of the averaged equations and Melnikov method. Therefore, we consider the averaged equations of the system subject to Duffing-Van der Pol s...     

An efficiently implementable Gauss-Newton-like method for solving singular nonlinear equations

A Gauss-Newton-like method for solving singular nonlinear equations is presented. The local convergence analysis shows that this method converges quadratically. The algorithm requires second derivative information in the formF″ ab only, which makes it attractive from the viewpoint of computational effort.     

求解自治非线性薛定谔方程的分离变量法

薛定谔方程是量子力学的基本方程,与经典物理中的牛顿运动方程地位相当.本文针对哈密顿量与时间无关的量子系统,应用分离变量法研究其量子力学定态解.分别给出了包含克尔型、饱和型以及五次非线性效应的薛定谔方程的定态解,并将所得解析解与数值解进行比较.两者完全吻合.     

Fixed point method for solving nonlinear quadratic Volterra integral equations

We consider a paper of Bana? and Sadarangani (2008) [11] which deals with monotonicity properties of the superposition operator and their applications. An application of the monotonicity properties is to study the solvability of a quadratic Volterra integral equation. In this paper, we prepare an efficient numerical technique based on the fixed point method and quadrature rules to approximate a solution for quadratic Volterra integral equation. Then convergence of numerical scheme is proved by some theorems and some numerical examples are given to show applicability and accuracy of the numerical method and guarantee the theoretical results.     

An approximation—Iteration method for solving nonlinear operator equations

We consider the problem of approximate determination of isolated bounded-norm solutions of nonlinear operator equations in a Hilbert space. Closed balls are constructed, such that the existence and uniqueness conditions are satisfied in each ball.Translated from Kibernetika, No. 1, pp. 21–28, January–February, 1991.     

The neural network collocation method for solving partial differential equations

Neural Computing and Applications - This paper presents a meshfree collocation method that uses deep learning to determine the basis functions as well as their corresponding weights. This method is...     

The modified differential transform method for solving MHD boundary-layer equations

A new analytical method (DTM-Padé) was developed for solving magnetohydrodynamic boundary-layer equations. It was shown that differential transform method (DTM) solutions are only valid for small values of independent variable. Therefore the DTM is not applicable for solving MHD boundary-layer equations, because in the boundary-layer problem y→∞. Numerical comparisons between the DTM-Padé and numerical methods (by using a fourth-order Runge-Kutta and shooting method) revealed that the new technique is a powerful method for solving MHD boundary-layer equations.     

An extrapolated splitting method for solving semi-discretized parabolic differential equations

An extrapolated Peaceman–Rachford–Strang splitting method is designed and examined in application to semi-discretized parabolic partial differential equations. A multi-product expansion method is implemented to improve the order of accuracy beyond second order in time. The numerical analysis of the high-order splitting method is further validated and illustrated through numerical experiments of linear and nonlinear partial differential equations.