Showing papers in "Mediterranean Journal of Mathematics in 2016"
TL;DR: This work constructs RKN methods from PRK methods with up to five stages and fourth algebraic order, andumerical results are given for the two-body problem and the perturbed two- body problem.
Abstract: In this work, we give the general framework for constructing trigonometrically fitted symplectic Runge–Kutta–Nystrom (RKN) methods from symplectic trigonometrically fitted partitioned Runge–Kutta (PRK) methods. We construct RKN methods from PRK methods with up to five stages and fourth algebraic order. Numerical results are given for the two-body problem and the perturbed two-body problem.
105 citations
TL;DR: In this paper, a mathematical approach is presented to identify the correct range of physical parameters adopting the recently popular analytic approximate Adomian decomposition method (ADM) up to a specified truncation order, the squared residual error formula is employed to work out the threshold and the existence domain of certain physical parameters satisfying a preassigned tolerance.
Abstract: Physical parameters in dimensionless form in the governing equations of real-life phenomena naturally occur. How to control them by determining their range of validity is in general a big issue. In this paper, a mathematical approach is presented to identify the correct range of physical parameters adopting the recently popular analytic approximate Adomian decomposition method (ADM). Having found the approximate analytical Adomian series solution up to a specified truncation order, the squared residual error formula is employed to work out the threshold and the existence domain of certain physical parameters satisfying a preassigned tolerance. If the current procedure is not closely pursued, the presented results with the ADM may not be up to the desired level of accuracy (the worst is the divergent physically meaningless solutions), or much more ADM series terms need to be computed to satisfy certain accuracy. Examples reveal the necessity of the present approach to make sure that the results embark the correct range of physical parameters in the study of a physical problem containing several dominating parameters.
104 citations
TL;DR: In this article, a four-stage high algebraic order symmetric two-step method with vanished phase-lag and its first up to the fourth derivative was developed, and the numerical results from the numerical tests are based on the numerical solution of the Schrodinger equation.
Abstract: In this paper, we will develop a four-stage high algebraic order symmetric two-step method with vanished phase-lag and its first up to the fourth derivative. For the proposed method, we will study the following: the phase-lag analysis of the new method; the development of the new method; the local truncation error analysis which is based on the radial Schrodinger equation; the stability and the interval of periodicity analysis which is based on a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis; the error estimation procedure which is based on the algebraic order; and the numerical results from our numerical tests for the examination of the efficiency of the new obtained method. The numerical tests are based on the numerical solution of the Schrodinger equation.
88 citations
TL;DR: In this article, the Ulam-Hyers stability of linear fractional differential equations using Laplace transform method was obtained as a direct corollary to Ulam's type stability of fractional evolution equations.
Abstract: In this paper, we first utilize fractional calculus, the properties of classical and generalized Mittag-Leffler functions to prove the Ulam–Hyers stability of linear fractional differential equations using Laplace transform method. Meanwhile, Ulam–Hyers–Rassias stability result is obtained as a direct corollary. Finally, we apply the same techniques to discuss the Ulam’s type stability of fractional evolution equations, impulsive fractional evolutions equations and Sobolev-type fractional evolution equations.
87 citations
TL;DR: Based on a suitable ansatz approach and Hirota's bilinear form, kink solitary wave, rogue wave and mixed exponential-algebraic solitary wave solutions of (2+1)-dimensional Burgers equation are derived as discussed by the authors.
Abstract: Based on a suitable ansatz approach and Hirota’s bilinear form, kink solitary wave, rogue wave and mixed exponential–algebraic solitary wave solutions of (2+1)-dimensional Burgers equation are derived. The completely non-elastic interaction between kink solitary wave and rogue wave for the (2+1)-dimensional Burgers equation are presented. These results enrich the variety of the dynamics of higher dimensional nonlinear wave field.
85 citations
TL;DR: In this paper, the authors considered a class of second-order evolution differential inclusions in Hilbert spaces and established sufficient conditions for the approximate controllability of such systems, and extended the results to non-local conditions.
Abstract: In this paper, we consider a class of second-order evolution differential inclusions in Hilbert spaces. This paper deals with the approximate controllability for a class of second-order control systems. First, we establish a set of sufficient conditions for the approximate controllability for a class of second-order evolution differential inclusions in Hilbert spaces. We use Bohnenblust–Karlin’s fixed point theorem to prove our main results. Further, we extend the result to study the approximate controllability concept with nonlocal conditions and also extend the result to study the approximate controllability for impulsive control systems with nonlocal conditions. An example is also given to illustrate our main results.
58 citations
TL;DR: In this article, the authors presented some numerical examples which support numerical results for the time fractional Burgers equation with various boundary and initial conditions obtained by collocation method using cubic B-spline base functions.
Abstract: We present some numerical examples which support numerical results for the time fractional Burgers equation with various boundary and initial conditions obtained by collocation method using cubic B-spline base functions. The aim of this paper is to show that the finite element method based on the cubic B-spline collocation method approach is also suitable for the treatment of the fractional differential equations. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme.
57 citations
TL;DR: In this article, the authors investigated numerical solutions of the regularized long wave (RLW) equation by using Haar wavelet (HW), combined with finite difference method, and the results of computations are compared with exact solutions and those already published.
Abstract: In this paper, we are going to investigate numerical solutions of the regularized long wave (RLW) equation by using Haar wavelet (HW), combined with finite difference method. The motion of a single solitary wave, interaction of two solitary waves, Maxwellian initial condition and wave undulation are our test problems for measuring performance of the proposed method. The results of computations are compared with exact solutions and those already published. \({L_{2}}\) and \({L_{\infty}}\) error norms and the numerical conservation laws are computed for discussing the accuracy and efficiency of the proposed method.
54 citations
TL;DR: In this article, the concept of hemi-slant submersion was introduced and the geometry of leaves of distributions which are involved in the definition of the sub-mersion were studied.
Abstract: As a generalization of anti-invariant submersions, semi-invariant submersions and slant submersions, we introduce the notion of hemi-slant submersion and study such submersions from Kahlerian manifolds onto Riemannian manifolds. After we study the geometry of leaves of distributions which are involved in the definition of the submersion, we obtain new conditions for such submersions to be harmonic and totally geodesic. Moreover, we give a characterization theorem for the proper hemi-slant submersions with totally umbilical fibers.
47 citations
TL;DR: In this article, the author found an explicit formula for the Bell numbers in terms of the Lah numbers and the Stirling numbers of the second kind, where the first kind is derived from the Lah number.
Abstract: In the paper, the author finds an explicit formula for the Bell numbers in terms of the Lah numbers and the Stirling numbers of the second kind.
44 citations
TL;DR: In this paper, the existence of mild solutions for a class of impulsive abstract differential equations with non-instantaneous impulses was studied and the Hausdorff measure of non-compactness was used.
Abstract: In this note we study the existence of global solutions for a class of impulsive abstract differential equations with non-instantaneous impulses. Specifically, we establish the existence of mild solutions on $${[0, \infty)}$$
and the existence of $${\mathcal{S}}$$
-asymptotically $${\omega}$$
-periodic mild solutions. Our results are based on the Hausdorff measure of non-compactness. Some applications involving partial differential equations are considered.
TL;DR: In this paper, the first eigenvalue corresponding to the relevant linear operator for the corresponding differential system is obtained, and the fixed point index of nonlinear operator in the Cartesian product space is calculated explicitly and the existence of at least one positive solution or two positive solutions of the singular differential system with integral boundary conditions is established.
Abstract: In this paper, by constructing a cone K
1 × K
2 in the Cartesian product space C[0, 1] × C[0, 1], and using spectral analysis of the relevant linear operator for the corresponding differential system, some properties of the first eigenvalue corresponding to the relevant linear operator are obtained, and the fixed-point index of nonlinear operator in the K
1 × K
2 is calculated explicitly and the existence of at least one positive solution or two positive solutions of the singular differential system with integral boundary conditions is established.
TL;DR: In this paper, the authors present sufficient conditions for approximate controllability of semilinear fractional control systems of order with infinite delay using strongly continuous cosine family and sequence method.
Abstract: The objective of this paper is to present some sufficient conditions for approximate controllability of semilinear fractional control system of order \({\alpha \in (1, 2]}\) with infinite delay. The results are obtained by the theory of strongly continuous \({\alpha}\)-order cosine family and sequence method. At the end, an example is given to illustrate the theory.
TL;DR: Aghajani et al. as mentioned in this paper introduced the concept of measure of non-compactness, and obtained a few generalizations of Darbo's fixed-point theorem.
Abstract: In the present paper, using the concept of measure of non-compactness, we introduce the concept of a new contraction on a Banach space and obtain few generalizations of Darbo’s fixed-point theorem and extend some recent results of (Aghajani et al., J. Comput. Appl. Math. 260:68–77, 2014) and (Aghajani et al., Bull. Belg. Math. Soc. Simon Stevin 20:345–358, 2013). Also we show the applicability of obtained results to the theory of integral equations. A concrete example illustrating the mentioned applicability is also included.
TL;DR: In this article, the authors considered a nonlinear wave equation with delay and showed that under suitable conditions on the initial data, the weights of the damping, the delay term and the nonlinear source, the energy of solutions blow up in a finite time.
Abstract: In this paper, we consider a nonlinear wave equation with delay. We show that under suitable conditions on the initial data, the weights of the damping, the delay term and the nonlinear source, the energy of solutions blows up in a finite time.
TL;DR: In this article, a comparison between a set-valued Gould type and simple Birkhoff integrals of bf(X)-valued multifunctions with respect to a non-negative set function is given.
Abstract: A comparison between a set-valued Gould type and simple Birkhoff integrals of bf(X)-valued multifunctions with respect to a non-negative set function is given. Relationships among them and Mc Shane multivalued integrability is given under suitable assumptions.
TL;DR: In this article, a new technique has been developed for analytical solutions of fractional order nonlinear ODE systems, called the fractional natural decomposition method (FNDM).
Abstract: A new technique has been developed for analytical solutions of fractional order nonlinear ODE system. We propose a reliable method called the fractional natural decomposition method (FNDM). The FNDM is based on the natural transform method (NTM) and the Adomian decomposition method. We use the FNDM to construct new analytical approximate and exact solutions to systems of nonlinear fractional ordinary differential equation (NLFODEs). The fractional derivatives are described in the Caputo sense.
TL;DR: In this paper, a computational method for solving multi-term differential equations based on new operational matrix of shifted second kind Chebyshev polynomials was proposed. But the convergence rate was not shown.
Abstract: This paper is devoted to studying a computational method for solving multi-term differential equations based on new operational matrix of shifted second kind Chebyshev polynomials. The properties of the operational matrix of fractional integration are exploited to reduce the main problem to an algebraic equation. We present an upper bound for the error in our estimation that leads to achieve the convergence rate of O(M
−κ
). Numerical experiments are reported to demonstrate the applicability and efficiency of the proposed method.
TL;DR: In this paper, the authors prove new dynamic Hardy-type inequalities on time scales with two different weighted functions, and apply the results to investigate the oscillation and nonoscillation of a half-linear second order dynamic equation on time scale.
Abstract: In this paper, we will prove some new dynamic Hardy-type inequalities on time scales with two different weighted functions. The study is to determine conditions on which the generalized inequalities hold using some known hypothesis. The main results will be proved by employing Holder’s inequality, Minkowski’s inequality and a chain rule on time scales. As special cases of our results, when the time scale is the real numbers, we will derive some well-known results due to Copson, Bliss, Flett and Bennett by a suitable choice of the weighted functions. We will apply the results to investigate the oscillation and nonoscillation of a half-linear second order dynamic equation on time scales.
TL;DR: In this article, a new class of generalized polynomials associated with the modified Milne-Thomson's polynomial is introduced, which is called the generalized Euler number.
Abstract: In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson’s polynomials $${\Phi_{n}^{(\alpha)}(x,
u)}$$
of degree n and order α introduced by Dere and Simsek. The concepts of Euler numbers E
n
, Euler polynomials E
n
(x), generalized Euler numbers E
n
(a, b), generalized Euler polynomials E
n
(x; a, b, c) of Luo et al., Hermite–Bernoulli polynomials $${{_HE}_n(x,y)}$$
of Dattoli et al. and $${{_HE}_n^{(\alpha)} (x,y)}$$
of Pathan are generalized to the one $${ {_HE}_n^{(\alpha)}(x,y,a,b,c)}$$
which is called the generalized polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between E
n
, E
n
(x), E
n
(a, b), E
n
(x; a, b, c) and $${{}_HE_n^{(\alpha)}(x,y;a,b,c)}$$
are established. Some implicit summation formulae and general symmetry identities are derived using different analytical means and applying generating functions.
TL;DR: In this article, the authors studied complex non-Kahler manifolds with Hermitian metrics being locally conformal to metrics with special cohomological properties, and provided examples where the existence of locally-conformal holomorphic-tamed structures implies the presence of locally conformally Kahler metrics, too.
Abstract: We study complex non-Kahler manifolds with Hermitian metrics being locally conformal to metrics with special cohomological properties. In particular, we provide examples where the existence of locally conformal holomorphic-tamed structures implies the existence of locally conformal Kahler metrics, too.
TL;DR: In this article, the existence and uniqueness of boundary value problems of fractional differential equations with p-Laplacian operator using fixed-point theorems in cone and coincidence degree theory were investigated.
Abstract: This paper deals with existence and uniqueness of solutions for several boundary value problems of fractional differential equations with p-Laplacian operator using fixed-point theorems in cone and coincidence degree theory. The main results enrich and extend some existing literatures. Some examples are given to illustrate our main results.
TL;DR: In this paper, a numerical method for solving time-fractional order telegraph equation is proposed based on an expansion of approximated solution in a series of Sinc function and shifted Legendre polynomials.
Abstract: In this paper, we introduce a numerical method for solving time-fractional order telegraph equation. The method depends basically on an expansion of approximated solution in a series of Sinc function and shifted Legendre polynomials. The fractional derivative is expressed in the Caputo definition of fractional derivatives. The expansion coefficients are then determined by reducing the time-fractional order telegraph equation with its boundary and initial conditions to a system of algebraic equations for these coefficients. This system can be solved numerically using the Newton’s iteration method. Several numerical examples are introduced to demonstrate the reliability and effectiveness of the introduced method.
TL;DR: A novel efficient matrix approach for solving the second-order linear matrix partial differential equations (MPDEs) under given initial conditions using two well-known subspace Krylov iterative methods.
Abstract: The basic aim of this article is to present a novel efficient matrix approach for solving the second-order linear matrix partial differential equations (MPDEs) under given initial conditions. For imposing the given initial conditions to the main MPDEs, the associated matrix integro-differential equations (MIDEs) with partial derivatives are obtained from direct integration with regard to the spatial variable x and time variable t. Hence, operational matrices of differentiation and integration together with the completeness of Bernoulli polynomials are used to reduce the obtained MIDEs to the corresponding algebraic Sylvester equations. Using two well-known subspace Krylov iterative methods (i.e., GMRES(10) and Bi-CGSTAB) we provide two algorithms for solving the mentioned Sylvester equations. A numerical example is provided to show the efficiency and accuracy of the presented approach.
TL;DR: In this paper, it was shown that the Lipschitz-free space over a separable ultrametric space has a monotone Schauder basis and is isomorphic to l 1.
Abstract: We prove that the Lipschitz-free space over a separable ultrametric space has a monotone Schauder basis and is isomorphic to l1. This extends results of A. Dalet using an alternative approach.
TL;DR: In this article, the impulsive boundary value problems for two classes of fractional differential equations with two different Caputo fractional derivatives and generalized boundary value conditions are considered, and sufficient conditions for existence and uniqueness of the solutions are established by applying well-known Banach's contraction mapping principle, Laplace transforms and some skills of inequalities.
Abstract: We consider the impulsive boundary value problems for two classes of fractional differential equations with two different Caputo fractional derivatives and generalized boundary value conditions Natural formulae of a solution for these problems are introduced, which can be regarded as a novelty item Some sufficient conditions for existence and uniqueness of the solutions to this nonlinear equations are established by applying well-known Banach’s contraction mapping principle, Laplace transforms and some skills of inequalities Finally, an example is given to illustrate the effectiveness of our results
TL;DR: In this paper, a modification on the Grunwald-Letnikov definition of fractional derivative was proposed, which consists of fixing the memory length and varying the lower terminal of the derivative.
Abstract: Contrary to integer-order derivative, the fractional-order derivative of a non-constant periodic function is not a periodic function with the same period. As a consequence of this property, the time-invariant fractional-order systems do not have any non-constant periodic solution unless the lower terminal of the derivative is ±∞, which is not practical. This property limits the applicability of the fractional derivative and makes it unfavorable, for a wide range of periodic real phenomena. Therefore, enlarging the applicability of fractional systems to such periodic real phenomena is an important research topic. In this paper, we give a solution for the above problem by imposing a simple modification on the Grunwald–Letnikov definition of fractional derivative. This modification consists of fixing the memory length and varying the lower terminal of the derivative. It is shown that the new proposed definition of fractional derivative preserves the periodicity.
TL;DR: In this paper, the existence of fractal interpolation function (FIF) for a sequence of data with countable iterated function system, where the integer order integral of FIF is revealed if the value of the integral is known at the initial endpoint or final endpoint.
Abstract: In recent years, the concept of fractal analysis is the best nonlinear tool towards understanding the complexities in nature. Especially, fractal interpolation has flexibility for approximation of nonlinear data obtained from the engineering and scientific experiments. Random fractals and attractors of some iterated function systems are more appropriate examples of the continuous everywhere and nowhere differentiable (highly irregular) functions, hence fractional calculus is a mathematical operator which best suits for analyzing such a function. The present study deals the existence of fractal interpolation function (FIF) for a sequence of data $${\{(x_n,y_n):n\geq 2\}}$$
with countable iterated function system, where $${x_n}$$
is a monotone and bounded sequence, $${y_n}$$
is a bounded sequence. The integer order integral of FIF for sequence of data is revealed if the value of the integral is known at the initial endpoint or final endpoint. Besides, Riemann–Liouville fractional calculus of fractal interpolation function had been investigated with numerical examples for analyzing the results.
TL;DR: The Ricci-flat Ricci solitons with the potential vector fields pointwise collinear with the Reeb vector fields on K-almost coKahler manifolds were studied in this article.
Abstract: Let M be a compact almost coKahler manifold. If the metric g of M is a Ricci soliton and the potential vector field is pointwise collinear with the Reeb vector field, then we prove that M is Ricci-flat and coKahler and the soliton g is steady. This generalizes a Goldberg-like conjecture for coKahler manifolds obtained by Cappelletti-Montano and Pastore, namely any compact Einstein K-almost coKahler manifold is coKahler. Without the assumption of compactness, Ricci solitons with the potential vector fields pointwise collinear with the Reeb vector fields on K-almost coKahler manifolds are also studied. Moreover, we prove that there exist no gradient Ricci solitons on proper \({(\kappa, \mu)}\)-almost coKahler manifolds.
TL;DR: In this article, the Hankel determinant was used to obtain coefficient expansions for Bi-Bazilevic functions in the open unit disk, where both the function and its inverse are Bazilevic there.
Abstract: A function is said to be bi-Bazilevic in the open unit disk U if both the function and its inverse are Bazilevic there. Making use of the Hankel determinant, in this work, we obtain coefficient expansions for Bi-Bazilevic functions.