The Runge–Kutta method for hybrid fuzzy differential equations

In this paper we study numerical methods for addressing hybrid fuzzy differential equations by an application of the Runge–Kutta method for fuzzy differential equations using the Seikkala derivative. We state a convergence result and give a numerical example to illustrate the theory.     

A fast singly diagonally implicit runge–kutta method for solving 1D unsteady convection‐diffusion equations

In this article, a fast singly diagonally implicit Runge–Kutta method is designed to solve unsteady one‐dimensional convection diffusion equations. We use a three point compact finite difference approximation for the spatial discretization and also a three‐stage singly diagonally implicit Runge–Kutta (RK) method for the temporal discretization. In particular, a formulation evaluating the boundary values assigned to the internal stages for the RK method is derived so that a phenomenon of the order of the reduction for the convergence does not occur. The proposed scheme not only has fourth‐order accuracy in both space and time variables but also is computationally efficient, requiring only a linear matrix solver for a tridiagonal matrix system. It is also shown that the proposed scheme is unconditionally stable and suitable for stiff problems. Several numerical examples are solved by the new scheme and the numerical efficiency and superiority of it are compared with the numerical results obtained by other methods in the literature. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 788–812, 2014     

Weak first- or second-order implicit Runge–Kutta methods for stochastic differential equations with a scalar Wiener process

New fully implicit stochastic Runge–Kutta schemes of weak order 1 or 2 are proposed for stochastic differential equations with sufficiently smooth drift and diffusion coefficients and a scalar Wiener process, which are derivative-free and which are A-stable in mean square for a linear test equation in some general settings. They are sought in a transparent way and their convergence order and stability properties are confirmed in numerical experiments.     

Generalized Kronecker product splitting iteration for the solution of implicit Runge–Kutta and boundary value methods

This paper is concerned with a generalization of the Kronecker product splitting (KPS) iteration for solving linear systems arising in implicit Runge–Kutta and boundary value methods discretizations of ordinary differential equations. It is shown that the new scheme can outperform the standard KPS method in some situations and can be used as an effective preconditioner for Krylov subspace methods. Numerical experiments are presented to demonstrate the effectiveness of the methods. Copyright © 2014 John Wiley & Sons, Ltd.     

The Hamilton–Waterloo problem for cycle sizes 3 and 4

The Hamilton–Waterloo problem seeks a resolvable decomposition of the complete graph K_n, or the complete graph minus a 1‐factor as appropriate, into cycles such that each resolution class contains only cycles of specified sizes. We completely solve the case in which the resolution classes are either all 3‐cycles or 4‐cycles, with a few possible exceptions when n=24 and 48. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 342–352, 2009     

Some new nonlinear wave solutions and their convergence for the (2+1)‐dimensional Broer–Kau–Kupershmidt equation

We use the bifurcation method of dynamical systems to study the (2+1)‐dimensional Broer–Kau–Kupershmidt equation. We obtain some new nonlinear wave solutions, which contain solitary wave solutions, blow‐up wave solutions, periodic smooth wave solutions, periodic blow‐up wave solutions, and kink wave solutions. When the initial value vary, we also show the convergence of certain solutions, such as the solitary wave solutions converge to the kink wave solutions and the periodic blow‐up wave solutions converge to the solitary wave solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

The convergence of fully discrete Galerkin approximations for the Benjamin–Bona–Mahony (BBM) equation

In the present paper, we analyze a second-order in time fully discrete finite element method for the BBM equation. The discretization in space is based on the standard Galerkin method, for the time discretization the Crank–Nicolson scheme is used. We also prove the convergence of a linearized Galerkin modification scheme.     

Necessary optimality conditions for fractional action‐like integrals of variational calculus with Riemann–Liouville derivatives of order (α, β)

We derive Euler–Lagrange‐type equations for fractional action‐like integrals of the calculus of variations which depend on the Riemann–Liouville derivatives of order (α, β), α>0, β>0, recently introduced by Cresson. Some interesting consequences are obtained and discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

Exact travelling wave solutions for the generalized nonlinear Schrödinger (GNLS) equation with a source by Extended tanh–coth, sine–cosine and Exp-Function methods

The capability of Extended tanh–coth, sine–cosine and Exp-Function methods as alternative approaches to obtain the analytic solution of different types of applied differential equations in engineering mathematics has been revealed. In this study, the generalized nonlinear Schrödinger (GNLS) equation is solved by three different methods. To obtain the single-soliton solutions for the equation, the Extended tanh–coth and sine–cosine methods are used. Furthermore, for this nonlinear evolution equation the Exp-Function method is applied to derive various travelling wave solution. Results show that while the first two procedures easily provide a concise solution, the Exp-Function method provides a powerful mathematical means for solving nonlinear evolution equations in mathematical physics.     

Hybrid,phase–fitted,four–step methods of seventh order for solving x″(t) = f(t,x)

A four‐step method of seventh algebraic order is presented. It is tuned for addressing the special second order initial value problem. The new method is hybrid, explicit, and uses three stages per step. In addition is phase fitted. In consequence it uses variable coefficients that depend on the magnitude of the step‐size. We also present numerical tests on a set of standard problems that illustrate the efficiency of the derived method over older ones given in the relevant literature.     

A numerical method based on extended Raviart–Thomas (ER‐T) mixed finite element method for solving damped Boussinesq equation

In this paper, we present the approximate solution of damped Boussinesq equation using extended Raviart–Thomas mixed finite element method. In this method, the numerical solution of this equation is obtained using triangular meshes. Also, for discretization in time direction, we use an implicit finite difference scheme. In addition, error estimation and stability analysis of both methods are shown. Finally, some numerical examples are considered to confirm the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Explicit exact solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, we construct new explicit exact solutions for the coupled the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation (KD equation) by using a improved mapping approach and variable separation method. By means of the method, new types of variable-separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) for the KD system are successfully obtained. The improved mapping approach and variable separation method can be applied to other higher-dimensional coupled nonlinear evolution equations.     

Wave solutions for variants of the KdV–Burger and the K(n,n)–Burger equations by the generalized G′/G‐expansion method

An application of the ‐expansion method to search for exact solutions of nonlinear partial differential equations is analyzed. This method is used for variants of the Korteweg–de Vries–Burger and the K(n,n)–Burger equations. The generalized ‐expansion method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact traveling wave solutions of nonlinear partial differential equations. It is shown that the generalized ‐expansion method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems. Copyright © 2017 John Wiley & Sons, Ltd.     

Computing the indices of Sturm–Liouville eigenvalues for coupled boundary conditions (the EIGENIND-SLP codes)

The main purpose of the EIGENIND-SLP codes is to compute the indices of known eigenvalues of self-adjoint Sturm–Liouville problems with coupled boundary conditions (BCs). The spectrum of the problems can be unbounded from both below and above. Using some recent theoretical results, the computation is converted to that of the indices of the same eigenvalues for appropriate separated BCs, and is then carried out in terms of the Prüfer angle. The algorithm so generated and its implementation are discussed, and numerous examples are presented to illustrate the theoretical results and various aspects of the implementation.     

Explicit hybrid six–step,sixth order,fully symmetric methods for solving y ″ = f (x,y)

A family of explicit, fully symmetric, sixth order, six‐step methods for the numerical solution of y′′ = f(x,y) is studied. This family wastes two function evaluations per step and can be derived through interpolation techniques. An interval of periodicity is possessed and the phase lag is of high order. Numerical instabilities usually present in such type of multistep methods were circumvented. We conclude with extended numerical tests over a set of problems justifying our effort of dealing with the new methods.     

Linear lower bounds for δc(p) for a class of 2D self‐destructive percolation models

The self‐destructive percolation model is defined as follows: Consider percolation with parameter p > p_c. Remove the infinite occupied cluster. Finally, give each vertex (or, for bond percolation, each edge) that at this stage is vacant, an extra chance δ to become occupied. Let δ_c(p) be the minimal value of δ, needed to obtain an infinite occupied cluster in the final configuration. This model was introduced by van den Berg and Brouwer. They showed, for the site model on the square lattice (and a few other 2D lattices satisfying a special technical condition) that δ_c(p) ≥ . In particular, δ_c(p) is at least linear in p ? p_c. Although the arguments used by van den Berg and Brouwer look very lattice‐specific, we show that they can be suitably modified to obtain similar linear lower bounds for δ_c(p) (with p near p_c) for a much larger class of 2D lattices, including bond percolation on the square and triangular lattices, and site percolation on the star lattice (or matching lattice) of the square lattice. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Generalized Steiner systems GS4 (2, 4, v,g) for g = 2, 3, 6

Generalized Steiner systems GS_d(t, k, v, g) were first introduced by Etzion and used to construct optimal constant‐weight codes over an alphabet of size g + 1 with minimum Hamming distance d, in which each codeword has length v and weight k. Much work has been done for the existence of generalized Steiner triple systems GS(2, 3, v, g). However, for block size four there is not much known on GS_d(2, 4, v, g). In this paper, the necessary conditions for the existence of a GS_d(t, k, v, g) are given, which answers an open problem of Etzion. Some singular indirect product constructions for GS_d(2, k, v, g) are also presented. By using both recursive and direct constructions, it is proved that the necessary conditions for the existence of a GS₄(2, 4, v, g) are also sufficient for g = 2, 3, 6. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 401–423, 2001     

Nonelement boundary representation with Bézier surface patches for 3D linear elasticity problems in parametric integral equation system (PIES) and its solving using Lagrange polynomials

In this article, we present a strategy of using rectangular and triangular Bézier surface patches for nonelement representation of 3D boundary geometries for problems of linear elasticity. The boundary generated in this way is directly incorporated in the parametric integral equation system (PIES), which has been developed by the authors. The boundary values on each surface patch are approximated by Lagrange polynomials. Three illustrative examples are presented to confirm the effectiveness of the proposed boundary representation in connection with PIES and to show good accuracy of numerical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 51–79, 2018     

Existence of Z-cyclic 3PDTWh(<Emphasis Type="Italic">p</Emphasis>) for Prime <Emphasis Type="Italic">p</Emphasis> ≡ 1 (mod 4)

A directed triplewhist tournament on p players over Z _p is said to have the three-person property if no two games in the tournament have three common players. We briefly denote such a design as a 3PDTWh(p). In this paper, we investigate the existence of a Z-cyclic 3PDTWh(p) for any prime p ≡ 1 (mod 4) and show that such a design exists whenever p ≡ 5, 9, 13 (mod 16) and p ≥ 29. This result is obtained by applying Weil’s theorem. In addition, we also prove that a Z-cyclic 3PDTWh(p) exists whenever p ≡ 1 (mod 16) and p < 10, 000 except possibly for p = 257, 769. Gennian Ge’s Research was supported by National Natural Science Foundation of China under Grant No. 10471127, Zhejiang Provincial Natural Science Foundation of China under Grant No. R604001, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.