Abstract: Abstract We consider the nonlinear Cauchy problem for \( \varPsi \) -Hilfer fractional differential equations and investigate the existence, interval of existence and uniqueness of solution in the weighted space of functions. The continuous dependence of solutions on initial conditions is proved via Weissinger fixed point theorem. Picard’s successive approximation method has been developed to solve nonlinear Cauchy problem for differential equations with \( \varPsi \) -Hilfer fractional derivative and an estimation has been obtained for the error bound. Further, by Picard’s successive approximation, we derive the representation formulae for the solution of linear Cauchy problem for \( \varPsi \) -Hilfer fractional differential equation with constant coefficient and variable coefficient in terms of Mittag-Leffler function and Generalized (Kilbas–Saigo) Mittag-Leffler function respectively. PubDate: 2019-03-22
Abstract: Abstract In this paper, we propose a relaxed nonlinear inexact Uzawa algorithm (RNIU) for solving the symmetric saddle point problems. It is an inner–outer iteration method with the inner iterations using variable accuracy for solving the approximate Schur complement system. The variable relaxation parameter is introduced to improve the convergence. We give the convergence analysis of this relaxed algorithm with variable inner accuracy, based on a simple energy norm. Sufficient conditions are given for the convergence of RNIU, which slightly improve the existing convergence results for the nonlinear inexact Uzawa algorithm with uniform inner accuracy in the literature. A practical approach for setting the variable relaxed parameters is proposed, and numerical experiments are given to illustrate the efficiency and sensitivity of RNIU. PubDate: 2019-03-21
Abstract: Abstract In this paper, we combine the theory of the reproducing kernel Hilbert spaces with the field of collocation methods to solve boundary value problems with a special emphasis on the reproducing property of kernels. Using the reproducing property of the kernels, a new efficient algorithm is proposed to obtain the cardinal functions of a reproducing kernel Hilbert space, which can be applied conveniently for multi-dimensional domains. The differentiation matrices are constructed and also a pointwise error estimate of applying them is derived. In addition, we prove the non-singularity of the collocation matrix. The proposed method is truly meshless, and can be applied conveniently and accurately for high order and also multi-dimensional problems. Numerical results are presented for the several problems such as second- and fifth-order two-point boundary value problems, one- and two-dimensional unsteady Burgers’ equations, and a three-dimensional parabolic partial differential equation. In addition, we compare the numerical results with the best-reported results in the literature to show the high accuracy and efficiency of the proposed method. PubDate: 2019-03-21
Abstract: Abstract In this paper, we treat a numerical scheme for the regular fractional Sturm–Liouville problem containing the Prabhakar fractional derivatives with the mixed boundary conditions. We show that the eigenfunctions corresponding to distinct numerical eigenvalues are orthogonal in the Hilbert spaces. The numerical errors and convergence rates are also investigated. Further, we consider a space-fractional diffusion equation and study the associated fractional Sturm–Liouville problem along with the convergence analysis. PubDate: 2019-03-21
Abstract: Abstract A bipolar fuzzy set is a powerful tool for depicting fuzziness and uncertainty. This model is more flexible and practical as compared to the fuzzy model. In this paper, we define certain notions, including a bipolar fuzzy number in the parametric form, the distance between two bipolar fuzzy number and bipolar fuzzy arithmetic. We illustrate these concepts with examples. We discuss the system of linear equations and its solution process with the right-hand side as parametric bipolar fuzzy numbers, and obtain the strong solution of the system. Further, we analyze our new approach to find the solutions of a fully bipolar fuzzy linear system of equations (FBFLSE) which is based on \((-1,1)\) -cut expansion. Moreover, by utilizing the proposed method, we determine the maximal and minimal symmetric solutions of the FBFLSEs which are based on a tolerable solution set and a controllable solution set, respectively. We also present numerical examples of our proposed FBFLSE. PubDate: 2019-03-21
Abstract: Abstract The aim of this paper is to present the notions of rough Pythagorean fuzzy ideals in semigroups. Then, this idea is extended to the lower and upper approximations of Pythagorean fuzzy left (resp. right) ideals, bi-ideals, interior ideals, \(\left( 1,2\right) \) -ideals in semigroups and some important properties related to these notions are given. PubDate: 2019-03-20
Abstract: Abstract In this paper, we present an alternative form of the correlated Stein–Stein option pricing model, which preserves the analytic tractability of the original form and also demonstrates certain advantage for model implementation in practice. We shall show, through an empirical study, that this alternative form generally outperforms the original Stein–Stein form for the selected set of data, demonstrating that it can serve as a good competitor to the original Stein–Stein form in real markets. PubDate: 2019-03-20
Abstract: Abstract The coupled system of time fractional derivatives of non-homogeneous Burgers’ equations is solved both analytically and numerically. The approximate analytical solutions of power series type are obtained by the fractional homotopy analysis transform method, which is verified numerically by the coupled fractional reduced differential transform method. Particularly, the present analytical solutions are compared with the results available in the literature for several special cases, and very excellent agreement was found. An error analysis is done to compute the average squared residual errors of those analytical approximations for other cases. The present analysis shows that the proposed technique has very excellent efficiency to give solutions with high precision. On the other hand, it is found that the optimal convergence control parameter derived here not only controls the convergence of the present solutions but also leads to identifying multiple solutions. PubDate: 2019-03-19
Abstract: Abstract In this paper, we introduce a semi-analytical method called the local fractional Laplace homotopy analysis method (LFLHAM) for solving wave equations with local fractional derivatives. The LFLHAM is based on the homotopy analysis method and the local fractional Laplace transform method, respectively. The proposed analytical method was a modification of the homotopy analysis method and converged rapidly within a few iterations. The nonzero convergence-control parameter was used to adjust the convergence of the series solutions. Three examples of non-differentiable wave equations were provided to demonstrate the efficiency and the high accuracy of the proposed technique. The results obtained were completely in agreement with the results in the existing methods and their qualitative and quantitative comparison of the results. PubDate: 2019-03-19
Abstract: Abstract In this study, first we derive a novel iteration scheme for the sign of a matrix with no pure imaginary eigenvalues. The fourth-order convergence speed of this scheme is given in detail. Secondly, we extend the obtained results so as to calculate the solution of the Yang–Baxter-like equation for the matrix A with no pure imaginary eigenvalues. Some numerical tests are also furnished to manifest the applicability of our method. PubDate: 2019-03-19
Abstract: Abstract In this paper, we investigate the initial value problems for a class of nonlinear fractional differential equations involving the variable-order fractional derivative. Our goal is to construct the spectral collocation scheme for the problem and carry out a rigorous error analysis of the proposed method. To reach this target, we first show that the variable-order fractional calculus of non constant functions does not have the properties like the constant order calculus. Second, we study the existence and uniqueness of exact solution for the problem using Banach’s fixed-point theorem and the Gronwall–Bellman lemma. Third, we employ the Legendre–Gauss and Jacobi–Gauss interpolations to conquer the influence of the nonlinear term and the variable-order fractional derivative. Accordingly, we construct the spectral collocation scheme and design the algorithm. We also establish priori error estimates for the proposed scheme in the function spaces \(L^{2}[0,1]\) and \(L^{\infty }[0,1]\) . Finally, numerical results are given to support the theoretical conclusions. PubDate: 2019-03-19
Abstract: Abstract This paper addresses modified-meshless numerical schemes for dynamical systems with proportional delays. The proposed mesh reduction techniques are based on a redial-point interpolation and moving least-squares methods. An optimal influence domain radius is constructed utilizing nodal connectivity and node-depending integration background mesh. Optimal shape parameters are obtained by the use of properties of the delta Kronecker and the compactly supported weight function. Numerical results are provided to justify the accuracy and efficiency of the proposed schemes. PubDate: 2019-03-19
Abstract: Abstract This study presents new Gauss’s inequalities for interval and fuzzy-interval-valued functions using the Aumann’s and Kaleva’s improper integrals. The inequality for interval-valued functions is based on the Kulisch–Miranker order relation. The order relation given in the fuzzy-interval space is defined level-wise through the Kulisch–Miranker order, and by means of this the inequality for fuzzy-interval-valued functions is interpreted. PubDate: 2019-03-18
Abstract: Abstract In this paper, we propose and analyze Chebyshev spectral collocation approximation for high-order nonlinear Volterra integro-differential equations. Under reasonable assumptions on the nonlinearity, it is shown that this numerical method converges exponentially in both \(L^\infty \) -norm and \(L^2\) -norm. Numerical results of several test examples are presented and comparisons are made with some existing numerical methods to prove the superiority and the effectiveness of the proposed method. PubDate: 2019-03-18
Abstract: Abstract A new family of conjugate gradient methods for large-scale unconstrained optimization problems is described. It is based on minimizing a penalty function, and uses a limited memory structure to exploit the useful information provided by the iterations. Our penalty function combines the good properties of the linear conjugate gradient method using some penalty parameters. We propose a suitable penalty parameter by solving an auxiliary linear optimization problem and show that the proposed parameter is promising. The global convergence of the new method is investigated under mild assumptions. Numerical results show that the new method is efficient and confirm the effectiveness of the memory structure. PubDate: 2019-03-18
Abstract: Abstract In this paper, we give new error bounds for linear complementarity problems when the involved matrices are S-Nekrasov matrices and B–S-Nekrasov matrices, respectively. We prove that the obtained bounds are better than those of Gao et al. (J Comput Appl Math 336:147–159, 2018) in some cases. PubDate: 2019-03-18
Abstract: Abstract In this paper, we establish sufficient conditions for the approximate solution mappings of parametric bilevel equilibrium problems with stability properties such as upper semicontinuity, lower semicontinuity, Hausdorff lower semicontinuity, continuity and Hausdorff continuity. Moreover, we also apply these results to parametric traffic network problems with equilibrium constraints. Many examples are provided to ensure the essentialness of the assumptions. PubDate: 2019-03-18
Abstract: Abstract In this paper, based on the multi-splitting iteration (MSI) method (Gu and Wang in J Appl Math Comput 42:479–490, 2013), we present a general multi-splitting iteration (GMSI) method for solving the PageRank problem. The convergence of the GMSI method is analyzed in detail. Moreover, the same idea can be used as a preconditioning technique to accelerate Krylov subspace methods, such as the GMRES method. Finally, several numerical examples are given to illustrate the effectiveness of the proposed algorithm. PubDate: 2019-03-18
Abstract: Abstract A high-order accurate compact scheme for the Swift–Hohenberg equation is presented in this paper. We discretize the Swift–Hohenberg equation by a fourth-order compact finite difference formula in space and a backward differentiation with second-order accurate in time, respectively. A stabilized splitting scheme is presented and a Newton-type iterative method is introduced to deal with the nonlinear term. Therefore, a large time step can be used. The resulting discrete systems are solved by a fast and efficient nonlinear multigrid solver. Adaptive time step method is implemented to reduce the computational cost. Various numerical simulations including a convergence test of the proposed scheme, comparison with second-order scheme, a test of the stability of the proposed scheme, extension of the adaptive time step method, comparison with the phase field crystal equation, a study of the effect of computational domain and boundary condition, and an evolution of Swift–Hohenberg equation in three dimensions, are performed to demonstrate the efficiency of our proposed method. PubDate: 2019-03-16
Abstract: Abstract In this work, the sphere-covering bound on covering codes in Rosenbloom–Tsfasman spaces (RT spaces) is improved by generalizing the excess counting method. The approach focuses on studying the parity of a Rosenbloom–Tsfasman sphere (RT sphere) and the parity of the intersection of two RT spheres. We connect the parity of an RT sphere with partial sums of binomial coefficients and p-adic valuation of binomial coefficients. The intersection number of RT spaces is introduced and we determinate its parity under some conditions. Numerical applications of the method are discussed. PubDate: 2019-03-16
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