共查询到18条相似文献,搜索用时 62 毫秒
1.
首次提出了一种分数阶差分,分数阶和分以及分数阶差分方程的定义,并给出(2,q)阶常系数分数阶差分方程的具体解法. 相似文献
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《数学的实践与认识》2020,(2)
主要研究了Riemann-Liouvile型和Caputo型分数阶差分方程解的存在唯一性.结合分数阶差分方程已有的研究理论,以向前差分为出发点,参考已有的向后差分的研究方法及相关的结论,推导出适用于向前差分方程的结论,运用相关结论及两个特殊函数的收敛性,证明了两类类分数阶差分方程解的存在唯一性.向后差分是就想要得到的目标状态推算出当前状态,而向前差分可以由目前状态推算出未来的目标状态.然而,迄今已有一系列以向后差分为出发点分数阶差分方程理论的专著问世,而鲜见以向前差分为出发点的分数阶差分方程理论.经过推导总结,得出了以向前差分为出发点的两类分数阶差分方程解的存在唯一性. 相似文献
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离散不等式,特别是离散的Gronwall不等式已被广泛应用于差分方程的研究.近年来,分数阶微分方程引起很多学者的关注.因此,利用一种新的分数阶和分的定义和不等式的方法,讨论一类更一般的离散分数阶Gronwall不等式. 相似文献
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考虑一类高阶分数阶差分方程边值问题.构造相关的格林函数,利用不等式技巧,分析格林函数的特征性质.运用不动点指数理论,获得了该分数阶差分方程边值问题存在多重正解的充分条件,举例说明了所获理论的有效性. 相似文献
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王金华向红军 《高校应用数学学报(A辑)》2018,(1):67-78
研究一类带p-Laplacian算子的分数阶差分方程边值问题.利用格林函数的特征性质、压缩映射原理及锥上的不动点定理等非线性方法,获得了该分数阶pLaplacian差分方程边值问题解的唯一性及正解的存在性条件,举例说明了所得结论的正确性. 相似文献
7.
时间分数阶扩散方程的数值解法 总被引:1,自引:0,他引:1
马亮亮 《数学的实践与认识》2013,43(10)
分数阶微分方程在许多应用科学上比整数阶微分方程更能准确地模拟自然现象.考虑时间分数阶扩散方程,将一阶的时间导数用分数阶导数α(0<α<1)替换,给出了一种计算有效的隐式差分格式,并证明了这个隐式差分格式是无条件稳定和无条件收敛的,最后用数值例子说明差分格式是有效的. 相似文献
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首先,把分数阶波方程转换成等价的积分-微分方程;然后,利用带权的分数阶矩形公式和紧差分算子分别对时间和空间方向进行离散.证明了当权重为1/2时,时间方向的收敛阶为α,其中α(1α2)为Caputo导数的阶数.利用Gronwall不等式,证明了数值格式的收敛性和稳定性.数值例子进一步表明了数值格式的有效性. 相似文献
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基于分数阶Taylor(泰勒)级数展开原理,建立单相延迟一阶分数阶近似方程,获得分数阶热传导方程.针对短脉冲激光加热问题建立分数阶热传导方程组,并运用Laplace(拉普拉斯)变换方法进行求解,给出非Gauss(高斯)时间分布的激光内热源温度场解析解.针对具体算例数值研究温度波传播特性.结果表明热传播速度与分数阶阶次有关,分数阶阶次增加,热传播速度减小,温度变化幅度增加.分数阶方程可以用于描述介于扩散方程和热波方程间的热传输过程,且对热传播机制与分数阶热传导方程中分数阶项的关系做了深入剖析. 相似文献
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Pham The Anh Artur Babiarz Adam Czornik Michal Niezabitowski Stefan Siegmund 《Mathematical Methods in the Applied Sciences》2020,43(13):7815-7824
In this paper, we establish some criteria for boundedness, stability properties, and separation of solutions of autonomous nonlinear nabla Riemann-Liouville scalar fractional difference equations. To derive these results, we prove the variation of constants formula for nabla Riemann-Liouville fractional difference equations. 相似文献
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Ahmet Bekir Özkan Güner Burcu Ayhan 《Mathematical Methods in the Applied Sciences》2015,38(17):3807-3817
In this paper, the ‐expansion method is proposed to establish hyperbolic and trigonometric function solutions for fractional differential‐difference equations with the modified Riemann–Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential‐difference equation into its differential‐difference equation of integer order. We obtain the hyperbolic and periodic function solutions of the nonlinear time‐fractional Toda lattice equations and relativistic Toda lattice system. The proposed method is more effective and powerful for obtaining exact solutions for nonlinear fractional differential–difference equations and systems. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
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Weinian Li Weihong Sheng Pingping Zhang 《Journal of Applied Analysis & Computation》2018,8(6):1910-1918
In this paper, we investigate the oscillation of a class of nonlinear fractional nabla difference equations. Some oscillation criteria are established. 相似文献
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On positive solutions to fractional sum boundary value problems for nonlinear fractional difference equations 下载免费PDF全文
Jiraporn Reunsumrit Thanin Sitthiwirattham 《Mathematical Methods in the Applied Sciences》2016,39(10):2737-2751
In this paper, we study a new class of 3‐point boundary value problems of nonlinear fractional difference equations. Our problems contain difference and fractional sum boundary conditions. Existence and uniqueness of solutions are proved by using the Banach fixed‐point theorem, and existence of the positive solutions is proved by using the Krasnoselskii's fixed‐point theorem. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
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Jinfa Cheng 《Annals of Differential Equations》2014,(1):5-14
This paper mainly discusses the problems of fractional variational problems and fractional diffusion problems using fractional difference and summation. Through the Euler finite difference method we get a variational formulation of the variation problem and the discrete solution to the time-fractional and space-fractional difference equation using separating variables method and two-side Z-transform method. 相似文献
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《Communications in Nonlinear Science & Numerical Simulation》2014,19(12):4057-4067
In this paper, we point out the differences between a class of fractional difference equations and the integer-order ones. We show that under the same boundary conditions, the problem of the fractional order is nonresonant, while the integer-order one is resonant. Then we analyse the discrete fractional boundary value problem in detail. Then the uniqueness and multiplicity of the solutions for the discrete fractional boundary value problem are obtained by two new tools established in 2012, respectively. 相似文献
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The transport dynamics in complex systems governing by anomalous diffusion modelled with Riesz fractional partial differential equations 下载免费PDF全文
Santanu Saha Ray 《Mathematical Methods in the Applied Sciences》2017,40(5):1637-1648
In this paper, numerical solutions of fractional Fokker–Planck equations with Riesz space fractional derivatives have been developed. Here, the fractional Fokker–Planck equations have been considered in a finite domain. In order to deal with the Riesz fractional derivative operator, shifted Grünwald approximation and fractional centred difference approaches have been used. The explicit finite difference method and Crank–Nicolson implicit method have been applied to obtain the numerical solutions of fractional diffusion equation and fractional Fokker–Planck equations, respectively. Numerical results are presented to demonstrate the accuracy and effectiveness of the proposed numerical solution techniques. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
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Numerical solutions of nonlinear fractional Schrödinger equations using nonstandard discretizations 下载免费PDF全文
Nasser H. Sweilam Taghreed A. Assiri Muner M. Abou Hasan 《Numerical Methods for Partial Differential Equations》2017,33(5):1399-1419
In this article, numerical study for both nonlinear space‐fractional Schrödinger equation and the coupled nonlinear space‐fractional Schrödinger system is presented. We offer here the weighted average nonstandard finite difference method (WANSFDM) as a novel numerical technique to study such kinds of partial differential equations. The space fractional derivative is described in the sense of the quantum Riesz‐Feller definition. Stability analysis of the proposed method is studied. To show that this method is reliable and computationally efficient different numerical examples are provided. We expect that the proposed schemes can be applicable to different systems of fractional partial differential equations. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1399–1419, 2017 相似文献