Convergence of the modified Halley’s method for multiple zeros under Hölder continuous derivative

In this paper, the estimate of the radius of the convergence ball of the modified Halley’s method for finding multiple zeros of nonlinear equations is provided under the hypotheses that the derivative f ^(m?+?1) of function f is Hölder continuous, and f ^(m?+?1) is bounded. The uniqueness ball of solution is also established. Finally, some examples are provided to show applications of our theorem.     

On the partial synchronization of iterative methods

The rapidly growing field of parallel computing systems promotes the study of parallel algorithms, with the Monte Carlo method and asynchronous iterations being among the most valuable types. These algorithms have a number of advantages. There is no need for a global time in a parallel system (no need for synchronization), and all computational resources are efficiently loaded (the minimum processor idle time). The method of partial synchronization of iterations for systems of equations was proposed by the authors earlier. In this article, this method is generalized to include the case of nonlinear equations of the form x = F(x), where x is an unknown column vector of length n, and F is an operator from ?ⁿ into ?ⁿ. We consider operators that do not satisfy conditions that are sufficient for the convergence of asynchronous iterations, with simple iterations still converging. In this case, one can specify such an incidence of the operator and such properties of the parallel system that asynchronous iterations fail to converge. Partial synchronization is one of the effective ways to solve this problem. An algorithm is proposed that guarantees the convergence of asynchronous iterations and the Monte Carlo method for the above class of operators. The rate of convergence of the algorithm is estimated. The results can prove useful for solving high-dimensional problems on multiprocessor computational systems.     

Unconditional and optimal <Emphasis Type="Bold">H</Emphasis><Superscript>2</Superscript>-error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schrödinger equation in high dimensions

The focus of this paper is on the optimal error bounds of two finite difference schemes for solving the d-dimensional (d = 2, 3) nonlinear Klein-Gordon-Schrödinger (KGS) equations. The proposed finite difference schemes not only conserve the mass and energy in the discrete level but also are efficient in practical computation because only two linear systems need to be solved at each time step. Besides the standard energy method, an induction argument as well as a ‘lifting’ technique are introduced to establish rigorously the optimal H ²-error estimates without any restrictions on the grid ratios, while the previous works either are not rigorous enough or often require certain restriction on the grid ratios. The convergence rates of the proposed schemes are proved to be at O(h ² + τ ²) with mesh-size h and time step τ in the discrete H ²-norm. The analysis method can be directly extended to other linear finite difference schemes for solving the KGS equations in high dimensions. Numerical results are reported to confirm the theoretical analysis for the proposed finite difference schemes.     

Dynamic study of Schröder’s families of first and second kind

The Schröder iterative families of the first and second kind are of great importance in the theory and practice of iterative processes for solving nonlinear equations f(x) = 0. In both cases, the methods E _r (first kind) and S _r (second kind) converge locally to a zero α of f as O(|x _k ? α| ^r ). Although characteristics of these families have been studied in many papers, their dynamic and chaotic behavior has not been completely investigated. In this paper, we compare convergence properties of both iterative schemes using the two methodologies: (i) comparison by numerical examples and (ii) comparison using dynamic study of methods by basins of attraction that enable their graphic visualization. Apart from the visualization of iterative processes, basins of attraction reveal very useful features on iterations such as consumed CPU time and average number of iterations, both as functions of starting points. We demonstrate by several examples that the Schröder family of the second kind S _r possesses better convergence characteristics than the Schröder family of the first kind E _r .     

Numerical approximation of solution for the coupled nonlinear Schrödinger equations

In this article, a compact finite difference scheme for the coupled nonlinear Schrödinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and convergence in maximum norm with order O(τ² + h⁴) are also proved by the discrete energy method. Finally, numerical results are provided to verify the theoretical analysis.     

Analysis of some numerical methods on layer adapted meshes for singularly perturbed quasilinear systems

We consider a coupled system of first-order singularly perturbed quasilinear differential equations with given initial conditions. The leading term of each equation is multiplied by a distinct small positive parameter, which induces overlapping layers. The quasilinear system is discretized by using first and second order accurate finite difference schemes for which we derive general error estimates in the discrete maximum norm. As consequences of these error estimates we establish nodal convergence of O((N ^?1 lnN) ^p ),p=1,2, on the Shishkin mesh and O(N ^?p ),p=1,2, on the Bakhvalov mesh, where N is the number of mesh intervals and the convergence is robust in all of the parameters. Numerical computations are included which confirm the theoretical results.     

Discrete nonlinear hyperbolic equations. Classification of integrable cases

We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on ?². The fields are associated with the vertices and an equation of the form Q(x ₁, x ₂, x ₃, x ₄) = 0 relates four vertices of one cell. The integrability of equations is understood as 3D-consistency, which means that it is possible to impose equations of the same type on all faces of a three-dimensional cube so that the resulting system will be consistent. This allows one to extend these equations also to the multidimensional lattices ? ^N . We classify integrable equations with complex fields x and polynomials Q multiaffine in all variables. Our method is based on the analysis of singular solutions.     

Maximal Regularity for Non-autonomous Equations with Measurable Dependence on Time

In this paper we study maximal L ^p -regularity for evolution equations with time-dependent operators A. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the L ^p -boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of m-th order elliptic operators A with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an L ^p (L ^q )-theory for such equations for \(p,q\in (1, \infty )\). In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.     

Non-polynomial spline method for the solution of two-dimensional linear wave equations with a nonlinear source term

In this paper, two classes of methods are developed for the solution of two space dimensional wave equations with a nonlinear source term. We have used non-polynomial cubic spline function approximations in both space directions. The methods involve some parameters, by suitable choices of the parameters, a new high accuracy three time level scheme of order O(h ⁴ + k ⁴ + τ ² + τ ² h ² + τ ² k ²) has been obtained. Stability analysis of the methods have been carried out. The results of some test problems are included to demonstrate the practical usefulness of the proposed methods. The numerical results for the solution of two dimensional sine-Gordon equation are compared with those already available in literature.     

Local Rigidity of Diophantine Translations in Higher-dimensional Tori

We prove a theorem asserting that, given a Diophantine rotation α in a torus T^d ≡ R^d/Z^d, any perturbation, small enough in the C^∞ topology, that does not destroy all orbits with rotation vector α is actually smoothly conjugate to the rigid rotation. The proof relies on a KAM scheme (named after Kolmogorov–Arnol’d–Moser), where at each step the existence of an invariant measure with rotation vector α assures that we can linearize the equations around the same rotation α. The proof of the convergence of the scheme is carried out in the C^∞ category.     

Recursive polynomial interpolation algorithm (RPIA)

Two difference schemes are derived for numerically solving the one-dimensional time distributed-order fractional wave equations. It is proved that the schemes are unconditionally stable and convergent in the \(L^{\infty }\) norm with the convergence orders O(τ ² + h ²+Δγ ²) and O(τ ² + h ⁴+Δγ ⁴), respectively, where τ,h, and Δγ are the step sizes in time, space, and distributed order. A numerical example is implemented to confirm the theoretical results.     

Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω R n be a bounded domain, H = L 2 (Ω), L : D(L) H → H be an unbounded linear operator, f ∈ C(■× R, R) and λ∈ R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = λf (x, u), u ∈ D(L), which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.     

Galerkin finite element method and error analysis for the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(τ+h^l+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(τ²+h^l+1).     

Finite element method for space-time fractional diffusion equation

In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order β (1 < β ≤ 2), and the first order time derivative by a Caputo fractional derivative of order γ (0 < γ ≤ 1). For the 0 < γ < 1 case, we present two schemes to approximate the time derivative and finite element methods for the space derivative, the optimal convergence rate can be reached O(τ^2?γ + h²) and O(τ² + h²), respectively, in which τ is the time step size and h is the space step size. And for the case γ = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(τ² + h²) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.     

Numerical solution to generalized Lyapunov/Stein and rational Riccati equations in stochastic control

We consider the numerical solution of the generalized Lyapunov and Stein equations in \(\mathbb {R}^{n}\), arising respectively from stochastic optimal control in continuous- and discrete-time. Generalizing the Smith method, our algorithms converge quadratically and have an O(n³) computational complexity per iteration and an O(n²) memory requirement. For large-scale problems, when the relevant matrix operators are “sparse”, our algorithm for generalized Stein (or Lyapunov) equations may achieve the complexity and memory requirement of O(n) (or similar to that of the solution of the linear systems associated with the sparse matrix operators). These efficient algorithms can be applied to Newton’s method for the solution of the rational Riccati equations. This contrasts favourably with the naive Newton algorithms of O(n⁶) complexity or the slower modified Newton’s methods of O(n³) complexity. The convergence and error analysis will be considered and numerical examples provided.     

Uniqueness of conservative solutions for nonlinear wave equations via characteristics

For some classes of one-dimensional nonlinear wave equations, solutions are Hölder continuous and the ODEs for characteristics admit multiple solutions. Introducing an additional conservation equation and a suitable set of transformed variables, one obtains a new ODE whose right hand side is either Lipschitz continuous or has directionally bounded variation. In this way, a unique characteristic can be singled out through each initial point. This approach yields the uniqueness of conservative solutions to various equations, including the Camassa-Holm and the variational wave equation u_tt ? c(u)(c(u)u_x )_x = 0, for general initial data in H¹(R).     

Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions

We present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of O(h) order in energy norm and of O(h²) order in L² norm on general d-rectangular triangulations. Moreover, when the triangulation is uniform, the convergence rate can be of O(h²) order in energy norm, and the convergence rate in L² norm is still of O(h²) order, which cannot be improved. Numerical examples are presented to demonstrate our theoretical results.     

A fractional spectral method with applications to some singular problems

In this paper we propose and analyze fractional spectral methods for a class of integro-differential equations and fractional differential equations. The proposed methods make new use of the classical fractional polynomials, also known as Müntz polynomials. We first develop a kind of fractional Jacobi polynomials as the approximating space, and derive basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We then construct efficient fractional spectral methods for some integro-differential equations which can achieve spectral accuracy for solutions with limited regularity. The main novelty of the proposed methods is that the exponential convergence can be attained for any solution u(x) with u(x ^1/λ ) being smooth, where λ is a real number between 0 and 1 and it is supposed that the problem is defined in the interval (0,1). This covers a large number of problems, including integro-differential equations with weakly singular kernels, fractional differential equations, and so on. A detailed convergence analysis is carried out, and several error estimates are established. Finally a series of numerical examples are provided to verify the efficiency of the methods.     

The defocusing energy-supercritical NLS in higher dimensions

We consider the defocusing nonlinear Schr?dinger equations iu_t +△u =|u|~(p_u) with p being an even integer in dimensions d≥ 5. We prove that an a priori bound of critical norm implies global well-posedness and scattering for the solution.     

A superconvergent local discontinuous Galerkin method for nonlinear two-point boundary-value problems

In this paper, we present and analyze a superconvergent and high order accurate local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form u ^″ = f (t, u), which arise in a wide variety of engineering applications. We prove the L ² stability of the LDG scheme and optimal L ² error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p +?1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. Moreover, we show that the derivatives of the LDG solutions are superconvergent with order p +?1 toward the derivatives of Gausss-Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order p +?3/2 toward Gauss-Radau projections of the exact solutions. Our computational results indicate that the observed numerical superconvergence rate is p +?2. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p ≥?1 and for the periodic, Dirichlet, and mixed boundary conditions. All proofs are valid under the hypotheses of the existence and uniqueness theorem for BVPs. Several numerical results are presented to validate the theoretical results.