We present convergence analysis of operator splitting methods applied to the nonlinear Rosenau–Burgers equation. The equation is first splitted into an unbounded linear part and a bounded nonlinear part and …
We present convergence analysis of operator splitting methods applied to the nonlinear Rosenau–Burgers equation. The equation is first splitted into an unbounded linear part and a bounded nonlinear part and then operator splitting methods of Lie‐Trotter and Strang type are applied to the equation. The local error bounds are obtained by using an approach based on the differential theory of operators in Banach space and error terms of one and two‐dimensional numerical quadratures via Lie commutator bounds. The global error estimates are obtained via a Lady Windermere's fan argument. Lastly, a numerical example is studied to confirm the expected convergence order.
In this paper, an operator splitting method is used to analyze nonlinear Benjamin-Bona-Mahony-type equations. We split the equation into an unbounded linear part and a bounded nonlinear part and then …
In this paper, an operator splitting method is used to analyze nonlinear Benjamin-Bona-Mahony-type equations. We split the equation into an unbounded linear part and a bounded nonlinear part and then Lie-Trotter splitting is applied to the equation. The local error bounds are obtained by using the approach based on the differential theory of operators in a Banach space and the quadrature error estimates via Lie commutator bounds. The global error estimate is obtained via Lady Windermere's fan argument. Finally, to confirm the expected convergence order, numerical examples are studied.
Quantum (q,h)-Bernstein bases and basic hypergeometric series are two seemingly unrelated mathematical entities. In this work, it is indicated that they are deeply interrelated theories. This new insight into two …
Quantum (q,h)-Bernstein bases and basic hypergeometric series are two seemingly unrelated mathematical entities. In this work, it is indicated that they are deeply interrelated theories. This new insight into two theories enables the provision of new proofs for two basic hypergeometric sums. The q-Chu-Vandermonde formula for basic hypergeometric series is proved by the partition of unity property for (q,h)-Bernstein bases, and the q-Pffaf-Saalschütz formula for basic hypergeometric series is proved by the Marsden identity for (q,h)-Bernstein bases.
Quantum (q,h)-Bernstein bases and basic hypergeometric series are two seemingly unrelated mathematical entities. In this work, it is indicated that they are deeply interrelated theories. This new insight into two …
Quantum (q,h)-Bernstein bases and basic hypergeometric series are two seemingly unrelated mathematical entities. In this work, it is indicated that they are deeply interrelated theories. This new insight into two theories enables the provision of new proofs for two basic hypergeometric sums. The q-Chu-Vandermonde formula for basic hypergeometric series is proved by the partition of unity property for (q,h)-Bernstein bases, and the q-Pffaf-Saalschütz formula for basic hypergeometric series is proved by the Marsden identity for (q,h)-Bernstein bases.
We present convergence analysis of operator splitting methods applied to the nonlinear Rosenau–Burgers equation. The equation is first splitted into an unbounded linear part and a bounded nonlinear part and …
We present convergence analysis of operator splitting methods applied to the nonlinear Rosenau–Burgers equation. The equation is first splitted into an unbounded linear part and a bounded nonlinear part and then operator splitting methods of Lie‐Trotter and Strang type are applied to the equation. The local error bounds are obtained by using an approach based on the differential theory of operators in Banach space and error terms of one and two‐dimensional numerical quadratures via Lie commutator bounds. The global error estimates are obtained via a Lady Windermere's fan argument. Lastly, a numerical example is studied to confirm the expected convergence order.
In this paper, an operator splitting method is used to analyze nonlinear Benjamin-Bona-Mahony-type equations. We split the equation into an unbounded linear part and a bounded nonlinear part and then …
In this paper, an operator splitting method is used to analyze nonlinear Benjamin-Bona-Mahony-type equations. We split the equation into an unbounded linear part and a bounded nonlinear part and then Lie-Trotter splitting is applied to the equation. The local error bounds are obtained by using the approach based on the differential theory of operators in a Banach space and the quadrature error estimates via Lie commutator bounds. The global error estimate is obtained via Lady Windermere's fan argument. Finally, to confirm the expected convergence order, numerical examples are studied.
We provide a new analytical approach to operator splitting for equations of the type $u_t=Au+B(u)$, where $A$ is a linear operator and $B$ is quadratic. A particular example is the …
We provide a new analytical approach to operator splitting for equations of the type $u_t=Au+B(u)$, where $A$ is a linear operator and $B$ is quadratic. A particular example is the Korteweg–de Vries (KdV) equation $u_t-u u_x+u_{xxx}=0$. We show that the Godunov and Strang splitting methods converge with the expected rates if the initial data are sufficiently regular.
We discuss numerical quadratures in one and two dimensions, which is followed by a discussion regarding the differentiation of general operators in Banach spaces. In addition, we discuss the standard …
We discuss numerical quadratures in one and two dimensions, which is followed by a discussion regarding the differentiation of general operators in Banach spaces. In addition, we discuss the standard and fractional Sobolev spaces, and prove several properties for these spaces. We show that the operator splitting methods of the Godunov type and Strang type applied to the viscous Burgers equation, and the Korteweg-de Vries (KdV) equation (and other equations), have the correct convergence in the Sobolev spaces. In the proofs we use the new framework originally introduced in [11]. We investigate the Godunov method and Strang method numerically for the viscous Burgers equation and the KdV equation, and present different numerical methods for the subequations from the splitting. We numerically check the convergence rates for the split step size, in addition with other aspects for the numerical methods. We find that the operator splitting methods work well numerically for the two equations. For the viscous Burgers equation, we find that several combination of numerical solvers for the subequations work well on the test problems, while we for the KdV equation find only one combination of numerical solvers which works well on all test problems.
We provide a new analytical approach to operator splitting for equations of the type $u_t=Au+u u_x$ where $A$ is a linear differential operator such that the equation is well-posed. Particular …
We provide a new analytical approach to operator splitting for equations of the type $u_t=Au+u u_x$ where $A$ is a linear differential operator such that the equation is well-posed. Particular examples include the viscous Burgers equation, the Korteweg–de Vries (KdV) equation, the Benney–Lin equation, and the Kawahara equation. We show that the Strang splitting method converges with the expected rate if the initial data are sufficiently regular. In particular, for the KdV equation we obtain second-order convergence in $H^r$ for initial data in $H^{r+5}$ with arbitrary $r\ge 1$.
Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind. Most of …
Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind. Most of the new material presented relates to the initial-value problem for the equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="1em" /> <mml:mspace width="1em" /> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , whose solution u ( x,t ) is considered in a class of real nonperiodic functions defined for ࢤ∞ < x < ∞, t ≥0. As an approximation derived for moderately long waves of small but finite amplitude in particular physical systems, this equation has the same formal justification as the Korteweg-de Vries equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> with which ( a ) is to be compared in various ways. It is contended that ( a ) is in important respects the preferable model, obviating certain problematical aspects of ( b ) and generally having more expedient mathematical properties. The paper divides into two parts where respectively the emphasis is on descriptive and on rigorous mathematics In §2 the origins and immediate properties of equations ( a ) and ( b ) are discussed in general terms, and the comparative shortcomings of ( b ) are reviewed. In the remainder of the paper (§§ 3,4) - which can be read independently Preceding discussion _ an exact theory of ( a ) is developed. In § 3 the existence of classical solutions is proved: and following our main result, theorem 1, several extensions and sidelights are presented. In § 4 solutions are shown to be unique, to depend continuously on their initial values, and also to depend continuously on forcing functions added to the right-hand side of ( a ). Thus the initial-value problem is confirmed to be classically well set in the Hadamard sense. In appendix 1 a generalization of ( a ) is considered, in which dispersive effects within a wide class are represented by an abstract pseudo-differential operator. The physical origins of such an equation are explained in the style of § 2, two examples are given deriving from definite physical problems, and an existence theory is outlined. In appendix 2 a technical fact used in § 3 is established.
This paper is concerned with a generalization of the classical Bernstein polynomials where the function is evaluated at intervals which are in geometric progression. It is shown that, when the …
This paper is concerned with a generalization of the classical Bernstein polynomials where the function is evaluated at intervals which are in geometric progression. It is shown that, when the function is convex, the generalized Bernstein polynomials B n are monotonic in n , as in the classical case.
We introduce a new technique, a three-level average linear-implicit finite difference method, for solving the Rosenau-Burgers equation. A second-order accuracy on both space and time numerical solution of the Rosenau-Burgers …
We introduce a new technique, a three-level average linear-implicit finite difference method, for solving the Rosenau-Burgers equation. A second-order accuracy on both space and time numerical solution of the Rosenau-Burgers equation is obtained using a five-point stencil. We prove the existence and uniqueness of the numerical solution. Moreover, the convergence and stability of the numerical solution are also shown. The numerical results show that our method improves the accuracy of the solution significantly.
We are concerned with the numerical solution obtained by splitting methods of certain parabolic partial differential equations. Splitting schemes of order higher than two with real coefficients necessarily involve negative …
We are concerned with the numerical solution obtained by splitting methods of certain parabolic partial differential equations. Splitting schemes of order higher than two with real coefficients necessarily involve negative coefficients. It has been demonstrated that this second-order barrier can be overcome by using splitting methods with complex-valued coefficients (with positive real parts). In this way, methods of orders $3$ to $14$ by using the SuzukiâYoshida triple (and quadruple) jump composition procedure have been explicitly built. Here we reconsider this technique and show that it is inherently bounded to order $14$ and clearly sub-optimal with respect to error constants. As an alternative, we solve directly the algebraic equations arising from the order conditions and construct methods of orders $6$ and $8$ that are the most accurate ones available at present time, even when low accuracies are desired. We also show that, in the general case, 14 is not an order barrier for splitting methods with complex coefficients with positive real part by building explicitly a method of order $16$ as a composition of methods of order 8.
We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an $H^4$-regular solution, a first-order error bound in the $H^1$ norm is shown …
We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an $H^4$-regular solution, a first-order error bound in the $H^1$ norm is shown and used to derive a second-order error bound in the $L_2$ norm. For the cubic Schrödinger equation with an $H^4$-regular solution, first-order convergence in the $H^2$ norm is used to obtain second-order convergence in the $L_2$ norm. Basic tools in the error analysis are Lie-commutator bounds for estimating the local error and $H^m$-conditional stability for error propagation, where $m=1$ for the Schrödinger-Poisson system and $m=2$ for the cubic Schrödinger equation.
The Schrödinger eigenvalue problem is solved with the imaginary time propagation technique. The separability of the Hamiltonian makes the problem suitable for the application of splitting methods. High order fractional …
The Schrödinger eigenvalue problem is solved with the imaginary time propagation technique. The separability of the Hamiltonian makes the problem suitable for the application of splitting methods. High order fractional time steps of order greater than two necessarily have negative steps and cannot be used for this class of diffusive problems. However, there exist methods which use fractional complex time steps with positive real parts which can be used with only a moderate increase in the computational cost. We analyze the performance of this class of schemes and propose new methods which outperform the existing ones in most cases. On the other hand, if the gradient of the potential is available, methods up to fourth order with real and positive coefficients exist. We also explore this case and propose new methods as well as sixth-order methods with complex coefficients. In particular, highly optimized sixth-order schemes for near integrable systems using positive real part complex coefficients with and without modified potentials are presented. A time-stepping variable order algorithm is proposed and numerical results show the enhanced efficiency of the new methods.
In this paper, an operator splitting method is used to analyze nonlinear Benjamin-Bona-Mahony-type equations. We split the equation into an unbounded linear part and a bounded nonlinear part and then …
In this paper, an operator splitting method is used to analyze nonlinear Benjamin-Bona-Mahony-type equations. We split the equation into an unbounded linear part and a bounded nonlinear part and then Lie-Trotter splitting is applied to the equation. The local error bounds are obtained by using the approach based on the differential theory of operators in a Banach space and the quadrature error estimates via Lie commutator bounds. The global error estimate is obtained via Lady Windermere's fan argument. Finally, to confirm the expected convergence order, numerical examples are studied.
By making use of divided differences, new proofs are presented for Dougall's summation theorem for well-poised 7F6-series and Whipple's transformation between well-poised 7F6-series and balanced 4F3-series. Several new summation formulae …
By making use of divided differences, new proofs are presented for Dougall's summation theorem for well-poised 7F6-series and Whipple's transformation between well-poised 7F6-series and balanced 4F3-series. Several new summation formulae of higher order are established, including three identities of Fox–Wright function and an extension of Karlsson–Minton formula.
Two seemingly disparate mathematical entities - quantum Bernstein bases and hypergeometric series - are revealed to be intimately related. The partition of unity property for quantum Bernstein bases is shown …
Two seemingly disparate mathematical entities - quantum Bernstein bases and hypergeometric series - are revealed to be intimately related. The partition of unity property for quantum Bernstein bases is shown to be equivalent to the Chu-Vandermonde formula for hypergeometric series, and the Marsden identity for quantum Bernstein bases is shown to be equivalent to the Pfaff-Saalsch?tz formula for hypergeometric series. The equivalence of the q-versions of these formulas and identities is also demonstrated.
By evaluating divided differences, we present new proofs for some terminating well-poised q-series identities and establish three new formulae for bibasic series, tribasic series and multibasic series.
By evaluating divided differences, we present new proofs for some terminating well-poised q-series identities and establish three new formulae for bibasic series, tribasic series and multibasic series.
The aim of this paper is to consider the time-decay properties of the solution for the Rosenau-Burgers equation in the form In particular, we prove some algebraic time decay rates …
The aim of this paper is to consider the time-decay properties of the solution for the Rosenau-Burgers equation in the form In particular, we prove some algebraic time decay rates of the solution within some spatial Sobolev spaces. The asymptotic stability the solution of the corresponding of linear equation is also obtained. To prove all of these, we make using of the method of Fourier transform together with the energy method.
Weutilize formulas for basic hypergeometric series to derive identities and formulas for negative degree q-Bernstein bases, including the Marsden identity, the partition of unity property, the monomial representation formula, the …
Weutilize formulas for basic hypergeometric series to derive identities and formulas for negative degree q-Bernstein bases, including the Marsden identity, the partition of unity property, the monomial representation formula, the reparametrization formula, and the degree reduction formula. We show that all these identities are just special forms of the q-analogue of Gauss? formula. We also provide a new proof for the q-analogue of Gauss? formula by using the Marsden identity for negative degree q-Bernstein bases together with the identity theorem for analytic functions.
Existence and uniqueness of numerical solutions are shown for the KdVlike Rosenau equation describing the dynamics of dense discrete systems. A priori bound and the error estimates as well as …
Existence and uniqueness of numerical solutions are shown for the KdVlike Rosenau equation describing the dynamics of dense discrete systems. A priori bound and the error estimates as well as conservation of energy of the finite difference approximate solutions are discussed with numerical examples.
Numerical solutions for generalized Rosenau equation are considered and two energy conservative finite difference schemes are proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence, …
Numerical solutions for generalized Rosenau equation are considered and two energy conservative finite difference schemes are proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence, and priori error estimate of the scheme are proved using energy method. Numerical results demonstrate that two schemes are efficient and reliable.
We present a convergence analysis for exponential splitting methods applied to linear evolution equations. Our main result states that the classical order of the splitting method is retained in a …
We present a convergence analysis for exponential splitting methods applied to linear evolution equations. Our main result states that the classical order of the splitting method is retained in a setting of unbounded operators, without requiring any additional order condition. This is achieved by basing the analysis on the abstract framework of (semi)groups. The convergence analysis also includes generalizations to splittings consisting of more than two operators, and to variable time steps. We conclude by illustrating that the abstract results are applicable in the context of the Schrödinger equation with an external magnetic field or with an unbounded potential.