309,759 research outputs found
Linear rational finite difference solution for solving first-order fredholm integro-differential equations
In this paper, we deal with the application of the linear rational finite difference (LRFD) method together with the first-order quadrature scheme to derive the first-order quadrature-rational finite difference approximation equation for first-order linear Fredholm integro-differential equations (FIDE). Derivation of this approximation equation, the linear system can be generated in which its coefficient matrix is large and dense. To make a comparison, the classical finite difference method (FD) based on the second-order central difference scheme is also presented. In numerical experiments, the maximum values of absolute errors of the numerical solutions obtained by both methods have been compared. Therefore, it can be concluded that the accuracy of numerical solutions for the quadrature-LRFD gives more accurate than the quadrature-FD method
A generalised finite difference scheme based on compact integrated radial basis function for flow in heterogeneous soils
In the present paper, we develop a generalised finite difference approach based on compact integrated radial basis function (CIRBF) stencils for solving highly nonlinear Richards equation governing fluid movement in heterogeneous soils. The proposed CIRBF scheme enjoys a high level of accuracy and a fast convergence rate with grid refinement owing to the combination of the integrated RBF approximation and compact approximation where the spatial derivatives are discretised in terms of the information of neighbouring nodes in a stencil. The CIRBF method is first verified through the solution of ordinary differential equations, 2-D Poisson equations and a Taylor-Green vortex. Numerical comparisons show that the CIRBF method outperforms some other methods in the literature. The CIRBF method in conjunction with a rational function transformation method and an adaptive time-stepping scheme is then applied to simulate 1-D and 2-D soil infiltrations effectively. The proposed solutions are more accurate and converge faster than those of the finite different method employed with a second-order central difference scheme. Additionally, the present scheme also takes less time to achieve target accuracy in comparison with the 1D-IRBF and HOC schemes
Refinement of SOR iterative method for the linear rational finite difference solution of second-order Fredholm Integro-differential equations
The primary objective of this paper is to develop the Refinement of Successive Over-Relaxation (RSOR) method based on a three-point linear rational finite difference-quadrature discretization scheme for the numerical solution of second-order linear Fredholm integro-differential equation (FIDE). Besides, to illuminate the superior performance of the proposed method, some numerical examples are presented and solved by implementing three approaches which are the Gauss-Seidel (GS), the Successive Over-Relaxation (SOR) and the RSOR methods. Lastly, through the comparison of the results, it is verified that the RSOR method is more effective than the other two methods, especially when considering the aspects of the number of iterations and running time
Quantized mirror curves and resummed WKB
Based on previous insights, we present an ansatz to obtain quantization
conditions and eigenfunctions for a family of difference equations which arise
from quantized mirror curves in the context of local mirror symmetry of toric
Calabi-Yau threefolds. It is a first principles construction, which yields
closed expressions for the quantization conditions and the eigenfunctions when
. The key ingredient is the modular duality structure
of the underlying quantum integrable system. We use our ansatz to write down
explicit results in some examples, which are successfully checked against
purely numerical results for both the spectrum and the eigenfunctions.
Concerning the quantization conditions, we also provide evidence that, in the
rational case, this method yields a resummation of conjectured quantization
conditions involving enumerative invariants of the underlying toric Calabi-Yau
threefold.Comment: 37 pages, 8 figures, 1 table; v2: minor corrections, more details
added in section
Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation
The Yablonskii-Vorob'ev polynomials , which are defined by a second
order bilinear differential-difference equation, provide rational solutions of
the Toda lattice. They are also polynomial tau-functions for the rational
solutions of the second Painlev\'{e} equation (). Here we define
two-variable polynomials on a lattice with spacing , by
considering rational solutions of the discrete time Toda lattice as introduced
by Suris. These polynomials are shown to have many properties that are
analogous to those of the Yablonskii-Vorob'ev polynomials, to which they reduce
when . They also provide rational solutions for a particular
discretisation of , namely the so called {\it alternate discrete}
, and this connection leads to an expression in terms of the Umemura
polynomials for the third Painlev\'{e} equation (). It is shown that
B\"{a}cklund transformation for the alternate discrete Painlev\'{e} equation is
a symplectic map, and the shift in time is also symplectic. Finally we present
a Lax pair for the alternate discrete , which recovers Jimbo and Miwa's
Lax pair for in the continuum limit .Comment: 23 pages, IOP style. Title changed, and connection with Umemura
polynomials adde
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