Numerical Algorithms最新文献

In this work, we investigate the sequence of monic q-Hermite I-Sobolev type orthogonal polynomials of higher-order, denoted by ({mathbb {H}_{n}(x;q)}_{nge 0}), which are orthogonal with respect to the following non-standard inner product involving q-differences:

where (lambda ) belongs to the set of positive real numbers, (mathscr {D}_{q}^{j}) denotes the j-th q -discrete analogue of the derivative operator, (q^jalpha in mathbb {R}backslash (-1,1)), and ((qx,-qx;q)_{infty }d_{q}(x)) denotes the orthogonality weight with its points of increase in a geometric progression. Connection formulas between these polynomials and standard q-Hermite I polynomials are deduced. The basic hypergeometric representation of (mathbb {H}_{n}(x;q)) is obtained. Moreover, for certain real values of (alpha ) satisfying the condition (q^jalpha in mathbb {R}backslash (-1,1)), we present results concerning the location of the zeros of (mathbb {H}_{n}(x;q)) and perform a comprehensive analysis of their asymptotic behavior as the parameter (lambda ) tends to infinity.

This paper introduces and analyzes some new highly efficient iterative procedures for approximating fixed points of contractive-type mappings. The stability, data dependence, strong convergence, and performance of the proposed schemes are addressed. Numerical examples demonstrate that the newly introduced schemes produce approximations of great accuracy and comparable to other similar robust schemes appeared in the literature. Nevertheless, all the schemes developed here are more efficient than other robust schemes used for comparison.

In this paper, we consider an improved iterative method for solving the monotone inclusion problem in the form of (0 in A(x) + D(x) + N_{C}(x)) in a real Hilbert space, where A is a maximally monotone operator, D and B are monotone and Lipschitz continuous, and C is the nonempty set of zeros of the operator B. We investigate the weak ergodic and strong convergence (when A is strongly monotone) of the iterates produced by our considered method. We show that the algorithmic scheme can also be applied to minimax problems. Furthermore, we discuss how to apply the method to the inclusion problem involving a finite sum of compositions of linear continuous operators by using the product space approach and employ it for convex minimization. Finally, we present a numerical experiment in TV-based image inpainting to validate the proposed theoretical theorem.

This work focuses on the numerical approximations of neutral stochastic delay differential equations with their drift and diffusion coefficients growing super-linearly with respect to both delay variables and state variables. Under generalized monotonicity conditions, we prove that the backward Euler method not only converges strongly in the mean square sense with order 1/2, but also inherit the mean square exponential stability of the original equations. As a byproduct, we obtain the same results on convergence rate and exponential stability of the backward Euler method for stochastic delay differential equations under generalized monotonicity conditions. These theoretical results are finally supported by several numerical experiments.

In this paper, we present a fully discrete finite difference scheme with efficient convolution of artificial boundary conditions for solving the Cauchy problem associated with the one-dimensional linearized Benjamin-Bona-Mahony equation. The scheme utilizes the Padé expansion of the square root function in the complex plane to implement the fast convolution, resulting in significant reduction of computational costs involved in the time convolution process. Moreover, the introduction of a constant damping term in the governing equations allows for convergence analysis under specific conditions. The theoretical analysis is complemented by numerical examples that illustrate the performance of the proposed numerical method.

The present study aims to introduce a numerical approach based on the hybrid of block-pulse functions (BPFs), Bernoulli polynomials (BPs), and hypergeometric function for analyzing a class of fractional variational problems (FVPs). The FVPs are made by the Caputo derivative sense. To analyze this problem, first, we create an approximate for the Riemann-Liouville fractional integral operator for BPFs and BPs of the fractional order. In this framework and using the Gauss-Legendre points, the main problem is converted into a system of algebraic equations. In the follow-up, an accurate upper bound is obtained and some theorems are established on the convergence analysis. Moreover, the computational order of convergence and solvability of the proposed approach are displayed and approximated theoretically and numerically. Meanwhile, the thrust of the proposed scheme is compared with other sophisticated examples in the literature, demonstrating that the process is accurate and efficient.

For solving horizontal linear complementarity problem (HLCP), we propose a general double-relaxation two-sweep modulus-based matrix splitting iteration method and a double-relaxation two-sweep modulus-based matrix splitting iteration method which contain a series of methods, by using different splittings. When the system matrices are (H_+)-matrices, we analyze convergence theory of these methods. Numerical examples in this paper illustrate that these methods are more efficient than modulus-based matrix splitting iteration method and general modulus-based matrix splitting iteration method.

In this note, we consider real nonsymmetric tridiagonal 2-Toeplitz matrices (textbf{B}_n). First we give the asymptotic spectral and singular value distribution of the whole matrix-sequence ({textbf{B}_n}_n), which is described via two eigenvalue functions of a (2times 2) matrix-valued symbol. In connection with the above findings, we provide a characterization of the eigenvalues and eigenvectors of real tridiagonal 2-Toeplitz matrices (textbf{B}_n) of even order, that can be turned into a numerical effective scheme for the computation of all the entries of (textbf{B}_n^l), n even and l positive and small compared to n. We recall that a corresponding eigenvalue decomposition for odd order tridiagonal 2-Toeplitz matrices was found previously, while, for even orders, an implicit formula for all the eigenvalues is obtained.

In this manuscript, we introduce a novel hybrid iteration process called the Picard–SP iteration process. We apply this new iteration process to approximate fixed points of generalized (alpha )–nonexpansive mappings. Convergence analysis of our newly proposed iteration process is discussed in the setting of uniformly convex Banach spaces and results are correlated with some other existing iteration processes. The dominance of the newly proposed iteration process is exhibited with the help of a new numerical example. In the end, the comparison of polynomiographs generated by other well-known iteration processes with our proposed iteration process has been presented to make a strong impression of our proposed iteration process.

We present an (L^2) norm convergence of the implicit-explicit BDF2 scheme with variable-step for the unsteady incompressible Navier-Stokes equations with an inf-sup stable FEM for the space discretization. Under a weak step-ratio constraint (0<r_k:=tau _k/tau _{k-1}<4.864), our error estimate is mesh-robust in the sense that it completely removes the possibly unbounded quantities, such as (Gamma _N=sum _{k=1}^{N-2}max {0,r_{k}-r_{k+2}}) and (Lambda _N=sum _{k=1}^{N-1}(|r_{k}-1|+|r_{k+1}-1|)) included in previous studies. In this analysis, we integrate our recent theoretical framework that employs discrete orthogonal convolution (DOC) kernels with an auxiliary Stokes problem to split the convergence analysis into two distinct parts. In the first part, we address intricate consistency error estimates for the velocity, pressure and nonlinear convection term. The resulting estimates allow us to utilize the conventional methodologies within the DOC framework to preserve spatial accuracy. In the second part, through the use of the DOC technique, we prove that the proposed variable-step BDF2 scheme is of second-order accuracy in time with respect to the (L^2) norm. Extensive numerical simulations coupled with an adaptive time-stepping algorithm are performed to show the accuracy and efficiency of the proposed variable-step method for the incompressible flows.