Liouvillian solutions of linear difference–differential equations

For a field k$k$ with an automorphism σ$\sigma$ and a derivation δ$\delta$, we introduce the notion of Liouvillian solutions of linear difference–differential systems {σ(Y)=AY,δ(Y)=BY}$\{\sigma \left(Y\right)=AY,\delta \left(Y\right)=BY\}$ over k$k$ and characterize the existence of Liouvillian solutions in terms of the Galois group of the systems. In the forthcoming paper, we will propose an algorithm for deciding if linear difference–differential systems of prime order have Liouvillian solutions.     

Collaborative applications over peer-to-peer systems–challenges and solutions

Emerging collaborative Peer-to-Peer (P2P) systems require discovery and utilization of diverse, multi-attribute, distributed, and dynamic groups of resources to achieve greater tasks beyond conventional file and processor cycle sharing. Collaborations involving application specific resources and dynamic quality of service goals are stressing current P2P architectures. Salient features and desirable characteristics of collaborative P2P systems are highlighted. Resource advertising, selecting, matching, and binding, the critical phases in these systems, and their associated challenges are reviewed using examples from distributed collaborative adaptive sensing systems, cloud computing, and mobile social networks. State-of-the-art resource discovery/aggregation solutions are compared with respect to their architecture, lookup overhead, load balancing, etc., to determine their ability to meet the goals and challenges of each critical phase. Incentives, trust, privacy, and security issues are also discussed, as they will ultimately determine the success of a collaborative P2P system. Open issues and research opportunities that are essential to achieve the true potential of collaborative P2P systems are discussed.     

Delay optimal control and viscosity solutions to associated Hamilton–Jacobi–Bellman equations

In this article, optimal control problems of differential equations with delays are investigated for which the associated Hamilton–Jacobi–Bellman (HJB) equations are nonlinear partial differential equations with delays. This type of HJB equation has not been previously studied and is difficult to solve because the state equations do not possess smoothing properties. We introduce a new notion of viscosity solutions and identify the value functional of the optimal control problems as the unique solution to the associated HJB equations. An analytical example is given as application.     

Non-uniqueness of solutions of the Hamilton–Jacobi–Bellman equation for time-average control

In control of diffusion processes a very useful instrument is the equation for optimal strategy and cost. For the version of infinite time horizon with time averaging this equation is much more complicated than for the version of finite time horizon, and even than for the version of infinite time horizon with discounting. In particular, the equation solution may be non-unique. This problem of non-uniqueness is researched in book of A. Arapostathis et al., 2012, for special models—near-monotone. The result received in the book is extended in the article to an important general case—models with restrictions in control which guarantee ergodicity of the process. Besides we correct the proofs from the book.     

New generalized method to construct new non-travelling wave solutions and travelling wave solutions of K–D equations

With the aid of computerized symbolic computation, we obtain new types of general solution of a first-order nonlinear ordinary differential equation with six degrees of freedom and devise a new generalized method and its algorithm, which can be used to construct more new exact solutions of general nonlinear differential equations. The (2+1)-dimensional K–D equation is chosen to illustrate our algorithm such that more families of new exact solutions are obtained, which contain non-travelling wave solutions and travelling wave solutions.     

Positive periodic solutions of neutral predator–prey model with Beddington–DeAngelis functional response

By using a continuation theorem based on coincidence degree theory, some new sufficient conditions are obtained for the existence of positive periodic solutions for a neutral predator–prey model with the Beddington–DeAngelis functional response.     

Symbolic computation of exact solutions for the compound KdV-Sawada–Kotera equation

The generalized F-expansion method is applied to construct the exact solutions of the compound KdV-Sawada–Kotera equation by the aid of the symbolic computation system Maple. Some new exact solutions which include Jacobi elliptic function solutions, soliton solutions and triangular periodic solutions are obtained via this method.     

The numerical simulation of periodic solutions for a predator–prey system

In this article, the existence and global stability of periodic solutions for a semi-ratio-dependent predator–prey system with Holling IV functional response and time delays are investigated. Using coincidence degree theory and Lyapunov method, sufficient conditions for the existence and global stability of periodic solutions are obtained. A numerical simulation is given to illustrate the results.     

Transformations and multi-solitonic solutions for a generalized variable-coefficient Kadomtsev–Petviashvili equation

Kadomtsev–Petviashvili equations with variable coefficients can be used to characterize many nonlinear phenomena in fluid dynamics and plasma physics more realistically than the equations with constant coefficients. Hereby, a generalized variable-coefficient Kadomtsev–Petviashvili equation with nonlinearity, dispersion and perturbed terms is investigated. Transformations, of which the consistency conditions are exactly the Painlevé integrability conditions, to the Korteweg–de Vries equation, cylindrical Korteweg–de Vries equation, Kadomtsev–Petviashvili equation and cylindrical Kadomtsev–Petviashvili equation are presented by formal dependent variable transformation assumptions. Using the Hirota bilinear method, from the variable-coefficient bilinear equation, the multi-solitonic solution, auto-Bäcklund transformation and Lax pair for the variable-coefficient Kadomtsev–Petviashvili equation are obtained. Moreover, the influence of inhomogeneity coefficients on solitonic structures and interaction properties is discussed for physical interest and possible applications.     

Numerical simulation of blow-up solutions for the generalized Davey–Stewartson system

Blow-up solutions for the generalized Davey–Stewartson system are studied numerically by using a split-step Fourier method. The numerical method has spectral-order accuracy in space and first-order accuracy in time. To evaluate the ability of the split-step Fourier method to detect blow-up, numerical simulations are conducted for several test problems, and the numerical results are compared with the analytical results available in the literature. Good agreement between the numerical and analytical results is observed.     

A framework for exploring numerical solutions of advection–reaction–diffusion equations using a GPU-based approach

In this paper we describe a general purpose, graphics processing unit (GP-GPU)-based approach for solving partial differential equations (PDEs) within advection–reaction–diffusion models. The GP-GPU-based approach provides a platform for solving PDEs in parallel and can thus significantly reduce solution times over traditional CPU implementations. This allows for a more efficient exploration of various advection–reaction–diffusion models, as well as, the parameters that govern them. Although the GPU does impose limitations on the size and accuracy of computations, the PDEs describing the advection–reaction–diffusion models of interest to us fit comfortably within these constraints. Furthermore, the GPU technology continues to rapidly increase in speed, memory, and precision, thus applying these techniques to larger systems should be possible in the future. We chose to solve the PDEs using two numerical approaches: for the diffusion, a first-order explicit forward Euler solution and a semi-implicit second order Crank–Nicholson solution; and, for the advection and reaction, a first-order explicit solution. The goal of this work is to provide motivation and guidance to the application scientist interested in exploring the use of the GP-GPU computational framework in the course of their research. In this paper, we present a rigorous comparison of our GPU-based advection–reaction–diffusion code model with a CPU-based analog, finding that the GPU model out-performs the CPU implementation in one-to-one comparisons.     

Lump solutions to the Kadomtsev–Petviashvili I equation with a self-consistent source

Based on symbolic computations, lump solutions to the Kadomtsev–Petviashvili I (KPI) equation with a self-consistent source (KPIESCS) are constructed by using the Hirota bilinear method and an ansatz technique. In contrast with lower-order lump solutions of the Kadomtsev–Petviashvili (KP) equation, the presented lump solutions to the KPIESCS exhibit more diverse nonlinear phenomena. The method used here is more natural and simpler.     

Solitary wave solutions for a generalized KdV–mKdV equation with variable coefficients

In this work, a generalized time-dependent variable coefficients combined KdV–mKdV (Gardner) equation arising in plasma physics and ocean dynamics is studied. By means of three amplitude ansatz that possess modified forms to those proposed by Wazwaz in 2007, we have obtained the bell type solitary waves, kink type solitary waves, and combined type solitary waves solutions for the considered model. Importantly, the results show that there exist combined solitary wave solutions in inhomogeneous KdV-typed systems, after proving their existence in the nonlinear Schrödinger systems. It should be noted that, the characteristics of the obtained solitary wave solutions have been expressed in terms of the time-dependent coefficients. Moreover, we give the formation conditions of the obtained solutions for the considered KdV–mKdV equation with variable coefficients.     

Bifurcation and exact traveling wave solutions for dual power Zakharov–Kuznetsov–Burgers equation with fractional temporal evolution

In this article, we introduce the dual power Zakharov–Kuznetsov–Burgers equation with fractional temporal evolution in the sense of modified Riemann–Liouville derivative. We investigate the dynamical behavior, bifurcations and phase portrait analysis of the exact traveling wave solutions of the system with and without damping effect. We apply the $(G^{\prime }∕G)$-expansion method in context of fractional complex transformation and seek a variety of exact traveling wave solutions including solitary wave, kink-type wave, breaking wave and periodic wave solutions of the equation. Furthermore, the remarkable features of the traveling wave solutions and phase portraits of dynamical system are demonstrated through interesting figures.     

Blow-up phenomena of solutions for a reaction–diffusion equation with weighted exponential nonlinearity

Blow-up phenomena for a reaction–diffusion equation with weighted exponential reaction term and null Dirichlet boundary condition are investigated. We establish sufficient conditions to guarantee existence of global solution or blow-up solution under appropriate measure sense by virtue of the method of super–sub solutions, the Bernoulli equation and the modified differential inequality techniques. Moreover, upper and lower bounds for the blow-up time are found in higher dimensional spaces and some examples for application are presented.     

Discretization methods with analytical solutions for a convection–reaction equation with higher-order discretizations

We introduce an improved second-order discretization method for the convection–reaction equation by combining analytical and numerical solutions. The method is derived from Godunov's scheme, see [S.K. Godunov, Difference methods for the numerical calculations of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. 47 (1959), pp. 271–306] and [R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.], and uses analytical solutions to solve the one-dimensional convection-reaction equation. We can also generalize the second-order methods for discontinuous solutions, because of the analytical test functions. One-dimensional solutions are used in the higher-dimensional solution of the numerical method.

The method is based on the flux-based characteristic methods and is an attractive alternative to the classical higher-order total variation diminishing methods, see [A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1993), pp. 357–393.]. In this article, we will focus on the derivation of analytical solutions embedded into a finite volume method, for general and special solutions of the characteristic methods.

For the analytical solution, we use the Laplace transformation to reduce the equation to an ordinary differential equation. With general initial conditions, e.g. spline functions, the Laplace transformation is accomplished with the help of numerical methods. The proposed discretization method skips the classical error between the convection and reaction equation by using the operator-splitting method.

At the end of the article, we illustrate the higher-order method for different benchmark problems. Finally, the method is shown to produce realistic results.     

An algorithm to compute Liouvillian solutions of prime order linear difference–differential equations

A normal form is given for integrable linear difference–differential equations {σ(Y)=AY,δ(Y)=BY}$\{\sigma \left(Y\right)=AY,\delta \left(Y\right)=BY\}$, which is irreducible over C(x,t)$\mathbb{C}(x,t)$ and solvable in terms of Liouvillian solutions. We refine this normal form and devise an algorithm to compute all Liouvillian solutions of such kinds of systems of prime order.     

A Neumann series of Bessel functions representation for solutions of Sturm–Liouville equations

A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm–Liouville equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter \(\omega \) the estimate of the difference between the exact solution and the approximate one (the truncated NSBF) depends on N (the truncation parameter) and the coefficients of the equation and does not depend on \(\omega \). A similar result is valid when \(\omega \in {\mathbb {C}}\) belongs to a strip \(\left| \hbox {Im }\omega \right| <C\). This feature makes the NSBF representation especially useful for applications requiring computation of solutions for large intervals of \(\omega \). Error and decay rate estimates are obtained. An algorithm for solving initial value, boundary value or spectral problems for the Sturm–Liouville equation is developed and illustrated on a test problem.