Exact solutions of a generalized (3 + 1)-dimensional Kadomtsev-Petviashvili equation using Lie symmetry analysis

In this paper, Lie group analysis is employed to derive some exact solutions of a generalized (3 + 1)-dimensional Kadomtsev-Petviashvili equation which describes the dynamics of solitons and nonlinear waves in plasmas and superfluids.     

Kawahara equation and modified Kawahara equation with time dependent coefficients: symmetry analysis and generalized ‐expansion method

In this paper, variable coefficients Kawahara equation (VCKE) and variable coefficients modified Kawahara equation (VCMKE), which arise in modeling of various physical phenomena, are studied by Lie group analysis. The similarity reductions and exact solutions are derived by determining the complete sets of point symmetries of these equations. Moreover, some exact analytic solutions are considered by the power series method. Further, a generalized ‐expansion method is applied to VCKE and VCMKE for constructing some new exact solutions. As a result, hyperbolic function solutions, trigonometric function solutions and some rational function solutions with parameters are furnished. Copyright © 2012 John Wiley & Sons, Ltd.     

λ‐symmetries,nonlocal transformations and first integrals to a class of Painlevé–Gambier equations

In this work, we consider a class of Painlevé–Gambier equations that model the motion of chain ball drawing with constant force in the frictionless surface. λ‐symmetries, first integrals, integrating factors, nonlocal transformations and local transformations are derived by using the some recent studies that are proposed by Muriel and Romero. Copyright © 2012 John Wiley & Sons, Ltd.     

A variety of distinct kinds of multiple soliton solutions for a ( 3 + 1)‐dimensional nonlinear evolution equation

In this work, a variety of distinct kinds of multiple soliton solutions is derived for a ( 3 + 1)‐dimensional nonlinear evolution equation. The simplified form of the Hirota's method is used to derive this set of distinct kinds of multiple soliton solutions. The coefficients of the spatial variables play a major role in the existence of this variety of multiple soliton solutions for the same equation. The resonance phenomenon is investigated as well. Copyright © 2012 John Wiley & Sons, Ltd.     

A note on the Painlevé analysis of a (2 + 1) dimensional Camassa–Holm equation

We investigate the Painlevé analysis for a (2 + 1) dimensional Camassa–Holm equation. Our results show that it admits only weak Painlevé expansions. This then confirms the limitations of the Painlevé test as a test for complete integrability when applied to non-semilinear partial differential equations.     

On a question of Thas on partial 3‐(qn + 1, q + 1, 1) designs

In this note, we answer a question of JA Thas about partial $3$‐ $\left(q^n+1,q+1,1\right)$ designs. We then extend this answer to a result about the embedding of certain partial $3$‐ $\left(q^2+1,q+1,1\right)$ designs into Möbius planes.     

A study on the (2 + 1)‐dimensional KdV4 equation derived by using the KdV recursion operator

We derive a new ( 2 + 1)‐dimensional Korteweg–de Vries 4 (KdV4) equation by using the recursion operator of the KdV equation. This study shows that the new KdV4 equation possess multiple soliton solutions the same as the multiple soliton solutions of the KdV hierarchy, but differ only in the dispersion relations. We also derive other traveling wave solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Two-dimensional symmetry reduction of (2 + 1)-dimensional nonlinear Klein–Gordon equation

In this paper, one-dimensional optimal system of group invariant solutions of (2 + 1)-dimensional Klein–Gordon system is constructed. Then the classification of group invariant solutions is given out and the corresponding two-dimensional symmetry reduced equations are obtained. At last some symmetry transformations are gained in detail. Especially, we obtain the most general solution from a given solution by use of six variable one-parameter subgroups transformations.     

More results on perfect (3,3,6k + 4; 6k − 2)‐threshold schemes

A (2,3)‐packing on X is a pair (X, ), where is a set of 3‐subsets (called blocks) of X, such that any pair of distinct points from X occurs together in at most one block. Its leave is a graph (X,E) such that E consists of all the pairs which do not appear in any block of . In this article, we shall construct a set of 6k ? 2 disjoint (2,3)‐packings of order 6k + 4 with K_1,3 ∪ 3kK₂ or G₁ ∪ (3k ? 1)K₂ as their common leave for any integer k ≥ 1 with a few possible exceptions (G₁ is a special graph of order 6). Such a system can be used to construct perfect threshold schemes as noted by Schellenberg and Stinson ( 22 ). © 2006 Wiley Periodicals, Inc. J Combin Designs     

A (3+1)‐dimensional Painlevé integrable model obtained by deformation

To find some non‐trivial higher‐dimensional integrable models (especially in (3+1) dimensions) is one of the most important problems in non‐linear physics. An efficient deformation method to obtain higher‐dimensional integrable models is proposed. Starting from (2+1)‐dimensional linear wave equation, a (3+1)‐dimensional non‐trivial non‐linear equation is obtained by using a non‐invertible deformation relation. Further, the Painlevé integrability of the resulting model is also proved. Copyright © 2002 John Wiley & Sons, Ltd.     

Comparison between homotopy analysis method and optimal homotopy asymptotic method for nth‐order integro‐differential equation

This paper presents general framework for solving the nth‐order integro‐differential equation using homotopy analysis method (HAM) and optimal homotopy asymptotic method (OHAM). OHAM is parameter free and can provide better accuracy over the HAM at the same order of approximation. Furthermore, in OHAM the convergence region can be easily adjusted and controlled. Comparison, via two examples, between our solution using HAM and OHAM and the exact solution shows that the HAM and the OHAM are effective and accurate in solving the nth‐order integro‐differential equation. Copyright © 2011 John Wiley & Sons, Ltd.     

A Lie symmetry analysis and explicit solutions of the two‐dimensional ∞‐Polylaplacian

In this work, we consider the Lie point symmetry analysis of a strongly nonlinear partial differential equation of third order, the ∞‐Polylaplacian, in two spatial dimensions. This equation is a higher order generalization of the ∞‐Laplacian, also known as Aronsson's equation, and arises as the analog of the Euler–Lagrange equations of a second‐order variational principle in L^∞. We obtain its full symmetry group, one‐dimensional Lie subalgebras and the corresponding symmetry reductions to ordinary differential equations. Finally, we use the Lie symmetries to construct new invariant ∞‐Polyharmonic functions.     

Three solutions for a class of quasilinear elliptic equation involving the p − q‐Laplace operator

The existence of at least three weak solutions is established for a class of quasilinear elliptic equation involving the p ? q‐Laplace operator with Dirichlet boundary condition. The technical approach is mainly on the basis of a three critical points theorem due to Ricceri. Copyright © 2013 John Wiley & Sons, Ltd.     

Group‐divisible designs with block size k having k + 1 groups,for k = 4, 5

We investigate the spectrum for k‐GDDs having k + 1 groups, where k = 4 or 5. We take advantage of new constructions introduced by R. S. Rees (Two new direct product‐type constructions for resolvable group‐divisible designs, J Combin Designs, 1 (1993), 15–26) to construct many new designs. For example, we show that a resolvable 4‐GDD of type g⁵ exists if and only if g ≡ 0 mod 12 and that a resolvable 5‐GDD of type g⁶ exists if and only if g ≡ 0 mod 20. We also show that a 4‐GDD of type g⁴m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 3 and 0 < m ≤ 3g/2, except possibly when (g,m) = (9,3) or (18,6), and that a 5‐GDD of type g⁵m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 4 and 0 < m ≤ 4g/3, with 32 possible exceptions. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 363–386, 2000     

A fourth‐order H1‐Galerkin mixed finite element method for Kuramoto–Sivashinsky equation

A H¹‐Galerkin mixed finite element method is applied to the Kuramoto–Sivashinsky equation by using a splitting technique, which results in a coupled system. The method described in this article may also be considered as a Petrov–Galerkin method with cubic spline space as trial space and piecewise linear space as test space, since the second derivative of a cubic spline is a linear spline. Optimal‐order error estimates are obtained without any restriction on the mesh for both semi‐discrete and fully discrete schemes. The advantage of this method over that presented in Manickam et al., Comput. Math. Appl. vol. 35(6) (1998) pp. 5–25; for the same problem is that the size (i.e., (n + 1) × (n + 1)) of each resulting linear system is less than half of the size of the linear system of the earlier method, where n is the number of subintervals in the partition. Further, there is a requirement of less regularity on exact solution in this method. The results are validated with numerical examples. Finally, instability behavior of the solution is numerically captured with this method.     

A symmetry classification for a class of (2+1)-nonlinear wave equation

In this paper, a symmetry classification of a (2+1)-nonlinear wave equation u_tt−f(u)(u_xx+u_yy)=0 where f(u) is a smooth function on u, using Lie group method, is given. The basic infinitesimal method for calculating symmetry groups is presented, and used to determine the general symmetry group of this (2+1)-nonlinear wave equation.     

A numerical method for the one‐dimensional sine‐Gordon equation

A numerical method based on a predictor–corrector (P‐C) scheme arising from the use of rational approximants of order 3 to the matrix‐exponential term in a three‐time level recurrence relation is applied successfully to the one‐dimensional sine‐Gordon equation, already known from the bibliography. In this P‐C scheme a modification in the corrector (MPC) has been proposed according to which the already evaluated corrected values are considered. The method, which uses as predictor an explicit finite‐difference scheme arising from the second order rational approximant and as corrector an implicit one, has been tested numerically on the single and the soliton doublets. Both the predictor and the corrector schemes are analyzed for local truncation error and stability. From the investigation of the numerical results and the comparison of them with other ones known from the bibliography it has been derived that the proposed P‐C/MPC schemes at least coincide in terms of accuracy with them. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Optimal (r,N)‐policy for discrete‐time Geo ∕ G ∕ 1 queue with different input rate and setup time

This paper presents a queue‐length analysis of Geo ∕ G ∕ 1 queue with ( r , N )‐policy and different input rate. Using a different method, the recursive expressions of queue‐length distribution at different epochs are obtained. Furthermore, some performance measures are also investigated. Finally, the Tabu search algorithm is used to search the joint optimum value of ( r , N ), which minimizes the state‐dependent operating cost. Copyright © 2014 John Wiley & Sons, Ltd.