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In this paper, a time‐fractional diffusion equation with singular source term is considered. The Caputo fractional derivative with order 0<α ?1 is applied to the temporal variable. Under specific initial and boundary conditions, we find that the time‐fractional diffusion equation presents quenching solution that is not globally well‐defined as time goes to infinity. The quenching time is estimated by using the eigenfunction of linear fractional diffusion equation. Moreover, by implementing a finite difference scheme, we give some numerical simulations to demonstrate the theoretical analysis. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
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Lung-An Ying. 《Mathematics of Computation》2004,73(246):595-611
We study a projection and upwind finite difference scheme for a combustion model problem. Convergence to weak solutions is proved under the Courant-Friedrichs-Lewy condition. More assumptions are given on the ignition temperature; then convergence to strong detonation wave solutions or to weak detonation wave solutions is proved.
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Ngoc Tran Vo Van Au Yong Zhou Nguyen Huy Tuan 《Mathematical Methods in the Applied Sciences》2020,43(6):3086-3098
In this paper, we study a backward problem for a fractional diffusion equation with nonlinear source in a bounded domain. By applying the properties of Mittag-Leffler functions and Banach fixed point theorem, we establish some results above the existence, uniqueness, and regularity of the mild solutions of the proposed problem in some suitable space. Moreover, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given, and the convergence rate between the regularized solution and the exact solution is also obtained. 相似文献
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YING Long'an 《中国科学A辑(英文版)》2004,47(Z1)
We study a finite difference scheme for a combustion model problem. A projection scheme near the combustion wave, and the standard upwind finite difference scheme away from the combustion wave are applied. Convergence to weak solutions with a combustion wave is proved under the normal Courant-Friedrichs-Lewy condition. Some con- 相似文献
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An initial value problem for the generalized Kolmogorov-Petrowsky-Piscunov (nonlinear degenerate reaction-diffusion) equation is studied numerically by the help of a slightly modified finite difference scheme of Douglas-Yanenko-Mimura type. If the initial function has compact support, the solution also will have compact support and an interface appears between the domains where the solution is positive and where it is zero. We present some examples for different parameter values where the numerical solution as well as numerical interfaces behave according to the analytical theory. This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
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提出了两个求解空间四阶的时间亚扩散方程的数值方法,其误差阶分别为O(τ+h2)和O(τ2+h2).通过Fourier方法,发现两个差分格式均为无条件稳定的.最后,通过数值例子,验证了两个算法的有效性. 相似文献
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A fully discrete spectral method for fractional Cattaneo equation based on Caputo–Fabrizo derivative
Haonan Li Shujuan Lü Tao Xu 《Numerical Methods for Partial Differential Equations》2019,35(3):936-954
Recently Caputo and Fabrizio introduced a new derivative with fractional order without singular kernel. The derivative can be used to describe the material heterogeneities and the fluctuations of different scales. In this article, we derived a new discretization of Caputo–Fabrizio derivative of order α (1 < α < 2) and applied it into the Cattaneo equation. A fully discrete scheme based on finite difference method in time and Legendre spectral approximation in space is proposed. The stability and convergence of the fully discrete scheme are rigorously established. The convergence rate of the fully discrete scheme in H1 norm is O(τ2 + N1?m), where τ, N and m are the time‐step size, polynomial degree and regularity in the space variable of the exact solution, respectively. Furthermore, the accuracy and applicability of the scheme are confirmed by numerical examples to support the theoretical results. 相似文献
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We establish necessary optimality conditions for variational problems with a Lagrangian depending on a combined Caputo derivative of variable fractional order. The endpoint of the integral is free, and thus transversality conditions are proved. Several particular cases are considered illustrating the new results. 相似文献
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一类非线性反应-扩散方程有限差分格式的稳定性研究 总被引:1,自引:0,他引:1
In the article,the fully discrete finite difference scheme for a type of nonlinear reaction-diffusion equation is established.Then the new function space is introduced and the stability problem for the finite difference scheme is discussed by means of variational approximation method in this function space.The approach used is of a simple characteristic in gaining the stability condition of the scheme. 相似文献
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Hossein Fazli HongGuang Sun Juan J. Nieto 《Mathematical Methods in the Applied Sciences》2022,45(1):197-205
We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy-type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well-known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory. 相似文献
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In this paper, a class of finite difference method for solving two-sided space-fractional wave equation is considered. The stability and consistency of the method are discussed by means of Gerschgorin theorem and using the stability matrix analysis. Numerical solutions of some wave fractional partial differential equation models are presented. The results obtained are compared to exact solutions. 相似文献
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In this paper, by introducing the fractional derivative in the sense of Caputo, the Adomian decomposition method is directly extended to study the coupled Burgers equations with time- and space-fractional derivatives. As a result, the realistic numerical solutions are obtained in a form of rapidly convergent series with easily computable components. The figures show the effectiveness and good accuracy of the proposed method. 相似文献
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淬火过程中具有非线性表面系数考虑相变时温度场的有限元分析 总被引:1,自引:0,他引:1
温度场的计算对淬火过程中热应力和热应变的分析有较大影响,对淬火后试件的残余应力和微观结构分析也有较大影响·本文以42CrMo钢圆柱体试件为研究对象,从相变等温动力学TTT图,给出了连续冷却相变动力学CCT图的数学模拟计算式,计算了钢淬火过程中相变组织成分百分比,将热物性系数处理为温度和相变体积百分比的函数,用有限单元法计算了淬火过程中具有非线性表面系数考虑相变时的温度场·并建立了相应的泛函,为以后计算淬火的热应力和热应变做好准备工作 相似文献
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Incorporating subdiffusive mechanisms into the Klein‐Kramers formalism leads to the fractional Klein‐Kramers equation. Then, the equation can effectively describe subdiffusion in the presence of an external force field in the phase space. This article presents the finite difference methods for numerically solving the fractional Klein‐Kramers equation and does the detailed stability and error analyses. The stability condition, mvβ ≤ 16, shows the ratio between the kinetic energy of the particle and the temperature of the fluid can not be too large, which well agrees with the physical property of the subdiffusive particle, we call it “physical constraint.” The numerical examples are provided to verify the theoretical results on rate of convergence. Moreover, we simulate the fractional Klein‐Kramers dynamics and the simulation results further confirm the effectiveness of our numerical schemes. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1561–1583, 2010 相似文献
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J. Alberto Conejero Carlos Lizama Ainara Mira-Iglesias Cristóbal Rodero 《Journal of Difference Equations and Applications》2019,25(9-10):1321-1331
ABSTRACTWe study time series generated by the parametric family of fractional discrete maps introduced by Wu and Baleanu, presenting an alternative way of introducing these maps. For the values of the parameters that yield chaotic time series, we have studied the Shannon entropy of the degree distribution of the natural and horizontal visibility graphs associated to these series. In these cases, the degree distribution can be fitted with a power law. We have also compared the Shannon entropy and the exponent of the power law fitting for the different values of the fractionary exponent and the scaling factor of the model. Our results illustrate a connection between the fractionary exponent and the scaling factor of the maps, with the respect to the onset of the chaos. 相似文献
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Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O(△t + h k+1 + H 2k+2 d/2 ) (k ≥ 1), where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems. 相似文献
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Vincent J. Ervin John Paul Roop 《Numerical Methods for Partial Differential Equations》2007,23(2):256-281
In this article, we discuss the steady state fractional advection dispersion equation (FADE) on bounded domains in ?d. Fractional differential and integral operators are defined and analyzed. Appropriate fractional derivative spaces are defined and shown to be equivalent to the fractional dimensional Sobolev spaces. A theoretical framework for the variational solution of the steady state FADE is presented. Existence and uniqueness results are proven, and error estimates obtained for the finite element approximation. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 256–281, 2007 相似文献