Nonlinear stochastic time-fractional slow and fast diffusion equations on Rd

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: $\left({?}^{\beta }+\frac{\nu }{2}{(?\Delta )}^{\alpha ∕2}\right)u(t,x)=I_t^{\gamma }\left[\rho \left(u(t,x)\right)\stackrel{?}{W}(t,x)\right],t>0,x\in {\mathbb{R}}^d,$where $\stackrel{?}{W}$ is the space–time white noise, $\alpha \in (0,2]$, $\beta \in (0,2)$, $\gamma \ge 0$ and $\nu >0$. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition: $d<2\alpha +\frac{\alpha }{\beta }min(2\gamma ?1,0)$. In some cases, the initial data can be measures. When $\beta \in (0,1]$, we prove the sample path regularity of the solution.     

A fast numerical method for solving coupled Burgers' equations

A new fast numerical scheme is proposed for solving time‐dependent coupled Burgers' equations. The idea of operator splitting is used to decompose the original problem into nonlinear pure convection subproblems and diffusion subproblems at each time step. Using Taylor's expansion, the nonlinearity in convection subproblems is explicitly treated by resolving a linear convection system with artificial inflow boundary conditions that can be independently solved. A multistep technique is proposed to rescue the possible instability caused by the explicit treatment of the convection system. Meanwhile, the diffusion subproblems are always self‐adjoint and coercive at each time step, and they can be efficiently solved by some existing preconditioned iterative solvers like the preconditioned conjugate galerkin method, and so forth. With the help of finite element discretization, all the major stiffness matrices remain invariant during the time marching process, which makes the present approach extremely fast for the time‐dependent nonlinear problems. Finally, several numerical examples are performed to verify the stability, convergence and performance of the new method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1823–1838, 2017     

A numerical method for an inhomogeneous nonlinear diffusion problem

A numerical method for the solution of an inhomogeneous nonlinear diffusion problem that arises in a variety of applications is presented. The diffusion coefficient in the underlying diffusion process is concentration- as well as distance- dependent. We wish to determine the concentration of the diffusing substance in a semi-infinite domain at any time, starting with a given initial concentration. The method of solution begins by first mapping the semi-infinite physical domain to a finite computational domain. An implicit finite-difference marching procedure is then used to advance the solution in time. Numerical results are presented for several physical problems. We observe that the present numerical solutions are in good agreement with the analytical solutions obtained previously by other researchers.     

Maximum principle and its application for the nonlinear time-fractional diffusion equations with Cauchy-Dirichlet conditions

In this paper, a maximum principle for the one-dimensional sub-diffusion equation with Atangana–Baleanu fractional derivative is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Atangana–Baleanu fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial–boundary-value problem for the linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.     

On M-Wright transforms and time-fractional diffusion equations

In this paper, we consider the Mittag-Leffler operator as an analytical solution of time-fractional diffusion equation in the Caputo sense. This solution is presented by an integral representation in terms of the M-Wright functions and the exponential operators. Further, we study the Mittag-Leffler operators associated with the Legendre and Bessel diffusion equations. Finally, we extend the obtained integral representation for the time-fractional diffusion equation of distributed order.     

A Kantorovich-type analysis for a fast iterative method for solving nonlinear equations

We revisit a fast iterative method studied by us in [I.K. Argyros, On a two-point Newton-like method of convergent order two, Int. J. Comput. Math. 88 (2) (2005) 219-234] to approximate solutions of nonlinear operator equations. The method uses only divided differences of order one and two function evaluations per step. This time we use a simpler Kantorovich-type analysis to establish the quadratic convergence of the method in the local as well as the semilocal case. Moreover we show that in some cases our method compares favorably, and can be used in cases where other methods using similar information cannot [S. Amat, S. Busquier, V.F. Candela, A class of quasi-Newton generalized Steffensen's methods on Banach spaces, J. Comput. Appl. Math. 149 (2) (2002) 397-406; D. Chen, On the convergence of a class of generalized Steffensen's iterative procedures and error analysis, Int. J. Comput. Math. 31 (1989) 195-203]. Numerical examples are provided to justify the theoretical results.     

A numerical method for finding soliton solutions in nonlinear differential equations

An iterative method for finding soliton solutions to Korteweg-de Vries and sine-Gordon equations, and for nonlinear Schrodinger equations with cubic nonlinearity, is proposed. In addition, the existence of soliton solutions depending on control parameters is investigated for femtosecond pulse propagation problems in media with cubic nonlinearity. In contrast to other approaches to finding soliton solutions, the proposed method has a high convergence rate, can be applied to multidimensional problems, and does not require special selection of the initial approximation.     

A numerical technique based on B-spline for a class of time-fractional diffusion equation

This paper presents an efficient numerical technique for solving a class of time-fractional diffusion equation. The time-fractional derivative is described in the Caputo form. The L1 scheme is used for discretization of Caputo fractional derivative and a collocation approach based on sextic B-spline basis function is employed for discretization of space variable. The unconditional stability of the fully-discrete scheme is analyzed. Two numerical examples are considered to demonstrate the accuracy and applicability of our scheme. The proposed scheme is shown to be sixth order accuracy with respect to space variable and (2 − α)-th order accuracy with respect to time variable, where α is the order of temporal fractional derivative. The numerical results obtained are compared with other existing numerical methods to justify the advantage of present method. The CPU time for the proposed scheme is provided.     

A linearized second-order scheme for nonlinear time fractional Klein-Gordon type equations

We consider difference schemes for nonlinear time fractional Klein-Gordon type equations in this paper. A linearized scheme is proposed to solve the problem. As a result, iterative method need not be employed. One of the main difficulties for the analysis is that certain weight averages of the approximated solutions are considered in the discretization and standard energy estimates cannot be applied directly. By introducing a new grid function, which further approximates the solution, and using ideas in some recent studies, we show that the method converges with second-order accuracy in time.     

A new regularization method for solving a time-fractional inverse diffusion problem

In this paper, we consider an inverse problem for a time-fractional diffusion equation in a one-dimensional semi-infinite domain. The temperature and heat flux are sought from a measured temperature history at a fixed location inside the body. We show that such problem is severely ill-posed and further apply a new regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under the a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

A numerical scheme based on a solution of nonlinear advection–diffusion equations

A mathematical formulation of the two-dimensional Cole–Hopf transformation is investigated in detail. By making use of the Cole–Hopf transformation, a nonlinear two-dimensional unsteady advection–diffusion equation is transformed into a linear equation, and the transformed equation is solved by the spectral method previously proposed by one of the authors. Thus a solution to initial value problems of nonlinear two-dimensional unsteady advection–diffusion equations is derived. On the base of the solution, a numerical scheme explicit with respect to time is presented for nonlinear advection–diffusion equations. Numerical experiments show that the present scheme possesses the total variation diminishing properties and gives solutions with good quality.     

A finite element method for the numerical solution of convection-dominated anisotropic diffusion equations

Summary. The proposed method is based on an additive decomposition of the differential operator and the subsequent fitted discretization of the resulting components. For standard situations, the derived stability and error estimates in the energy norm qualitatively coincide with well-known estimates. In the case of small diffusion, a uniform error estimate with reduced order is obtained. Received August 7, 1997 / Revised version received July 15, 1998 / Published online December 6, 1999     

A fast numerical method for block lower triangular Toeplitz with dense Toeplitz blocks system with applications to time-space fractional diffusion equations

Based on the circulant-and-skew-circulant representation of Toeplitz matrix inversion and the divide-and-conquer technique, a fast numerical method is developed for solving N-by-N block lower triangular Toeplitz with M-by-M dense Toeplitz blocks system with \(\mathcal {O}(MN\log N(\log N+\log M))\) complexity and \(\mathcal {O}(NM)\) storage. Moreover, the method is employed for solving the linear system that arises from compact finite difference scheme for time-space fractional diffusion equations with significant speedup. Numerical examples are given to show the efficiency of the proposed method.     

A novel numerical method and its convergence for nonlinear delay Volterra integro-differential equations

In this paper, we suggest a convergent numerical method for solving nonlinear delay Volterra integro-differential equations. First, we convert the problem into a continuous-time optimization problem and then use a shifted pseudospectral method to discrete the problem. Having solved the last problem, we can achieve the pointwise and continuous approximate solutions for the main delay Volterra integro-differential equations. Here, we analyze the convergence of the method and solve some numerical examples to show the efficiency of the method.     

A numerical method for branch points of a system of nonlinear algebraic equations

A numerical method is proposed for determining the branch points of a system of nonlinear algebraic equations which depend on a parameter. The basic idea of the present method is to replace a part of the original equations in the neighborhood of the branch point by a set of new equations until the point is reduced to a regular point, and hence its accurate point can easily be obtained with second-order convergence. An illustrative example is solved to show the effectiveness of the present algorithm.     

A fast sweeping method for Eikonal equations

In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses Gauss-Seidel iterations with alternating sweeping ordering to solve the discretized system. The crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding Eikonal equation in a certain direction simultaneously. The method has an optimal complexity of for grid points and is extremely simple to implement in any number of dimensions. Monotonicity and stability properties of the fast sweeping algorithm are proven. Convergence and error estimates of the algorithm for computing the distance function is studied in detail. It is shown that Gauss-Seidel iterations is enough for the distance function in dimensions. An estimation of the number of iterations for general Eikonal equations is also studied. Numerical examples are used to verify the analysis.

     

A linearized difference scheme for semilinear parabolic equations with nonlinear absorbing boundary conditions

A novel three level linearized difference scheme is proposed for the semilinear parabolic equation with nonlinear absorbing boundary conditions. The solution of this problem will blow up in finite time. Hence this difference scheme is coupled with an adaptive time step size, i.e., when the solution tends to infinity, the time step size will be smaller and smaller. Furthermore, the solvability, stability and convergence of the difference scheme are proved by the energy method. Numerical experiments are also given to demonstrate the theoretical second order convergence both in time and in space in L_∞-norm.