Search

In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time-fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs. We employ the time-stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation. Copyright © 2015 John Wiley & Sons, Ltd.

This paper is concerned with the optimal distributed control of the viscous weakly dispersive Degasperis–Procesi equation in nonlinear shallow water dynamics. It is well known that the Pontryagin maximum principle, which unifies calculus of variations and control theory of ordinary differential equations, sets up the theoretical basis of the modern optimal control theory along with the Bellman dynamic programming principle. In this paper, we commit ourselves to infinite dimensional generalizations of the maximum principle and aim at the optimal control theory of partial differential equations. In contrast to the finite dimensional setting, the maximum principle for the infinite dimensional system does not generally hold as a necessary condition for optimal control. By the Dubovitskii and Milyutin functional analytical approach, we prove the Pontryagin maximum principle of the controlled viscous weakly dispersive Degasperis–Procesi equation. The necessary optimality condition is established for the problem in fixed final horizon case. Finally, a remark on how to utilize the obtained results is also made. Copyright © 2015 JohnWiley & Sons, Ltd.

We present a high-order spectral element method (SEM) using modal (or hierarchical) basis for modeling of some nonlinear second-order partial differential equations in two-dimensional spatial space. The discretization is based on the conforming spectral element technique in space and the semi-implicit or the explicit finite difference formula in time. Unlike the nodal SEM, which is based on the Lagrange polynomials associated with the Gauss–Lobatto–Legendre or Chebyshev quadrature nodes, the Lobatto polynomials are used in this paper as modal basis. Using modal bases due to their orthogonal properties enables us to exactly obtain the elemental matrices provided that the element-wise mapping has the constant Jacobian. The difficulty of implementation of modal approximations for nonlinear problems is treated in this paper by expanding the nonlinear terms in the weak form of differential equations in terms of the Lobatto polynomials on each element using the fast Fourier transform (FFT). Utilization of the Fourier interpolation on equidistant points in the FFT algorithm and the enough polynomial order of approximation of the nonlinear terms can lead to minimize the aliasing error. Also, this approach leads to finding numerical solution of a nonlinear differential equation through solving a system of linear algebraic equations. Numerical results for some famous nonlinear equations illustrate efficiency, stability and convergence properties of the approximation scheme, which is exponential in space and up to third-order in time. Copyright © 2014 John Wiley & Sons, Ltd.

The aim of this paper is to deal with the existence of mild solutions and exact controllability for a class of fractional evolution inclusions with damping (FEID, for short) in Banach spaces. Firstly, we provide the representation of mild solutions for FEID by applying the method of Laplace transform and the theory of (α,κ)-regularized families of operators. Next, we are concerned with the existence and exact controllability of FEID under some suitable sufficient conditions by using the method of measure of noncompactness and an appropraite fixed point theorem. Finally, an application to nonlinear partial differential equations with temporal fractional derivatives is presented to illustrate the effectiveness of our main results. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider a variable coefficient Calogero–Degasperis equation, a variable coefficient potential Kadomstev–Petviashvili equation and the generalized (3+1)-dimensional variable coefficient Kadomtsev–Petviashvili equation with time-dependent coefficients. Shock wave solutions for the considered models are obtained by using ansatz method in the form of tanhp function. The physical parameters in the soliton solutions are obtained as functions of the dependent coefficients. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we apply the dual reciprocity boundary elements method for the numerical solution of two-dimensional linear and nonlinear time-fractional modified anomalous subdiffusion equations and time-fractional convection–diffusion equation. The fractional derivative of problems is described in the Riemann–Liouville and Caputo senses. We employ the linear radial basis function for interpolation of the nonlinear, inhomogeneous and time derivative terms. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity which appears in the nonlinear problems under consideration. The accuracy and efficiency of the proposed schemes are checked by five test problems. The proposed method is employed for solving some examples in two dimensions on unit square and also in complex regions to demonstrate the efficiency of the new technique. Copyright © 2016 John Wiley & Sons, Ltd.

We prove the existence of weak solutions to a one-dimensional initial-boundary value problem for a model system of partial differential equations, which consists of a sub-system of linear elasticity and a nonlinear non-uniformly parabolic equation of second order. To simplify the existence proof of weak solutions in the 2006 paper of Alber and Zhu, we replace the function |p|κ:=|p|2/|p|2+κ2 in that work by |p|κ:=|p|2+κ2. The model is formulated by using a sharp interface model for phase transformations that are driven by material forces. Copyright © 2017 John Wiley & Sons, Ltd.

The pseudo-spectral Legendre–Galerkin method (PS-LGM) is applied to solve a nonlinear partial integro-differential equation arising in population dynamics. This equation is a competition model in which similar individuals are competing for the same resources. It is a kind of reaction–diffusion equation with integral term corresponding to nonlocal consumption of resources. The proposed method is based on the Legendre–Galerkin formulation for the linear terms and interpolation operator at the Chebyshev–Gauss–Lobatto (CGL) points for the nonlinear terms. Also, the integral term, which is a kind of convolution, is directly computed by a fast and accurate method based on CGL interpolation operator, and thus, the use of any quadrature formula in its computation is avoided. The main difference of the PS-LGM presented in the current paper with the classic LGM is in treating the nonlinear terms and imposing boundary conditions. Indeed, in the PS-LGM, the nonlinear terms are efficiently handled using the CGL points, and also the boundary conditions are imposed strongly as collocation methods. Combination of the PS-LGM with a semi-implicit time integration method such as second-order backward differentiation formula and Adams-Bashforth method leads to reducing the complexity of computations and obtaining a linear algebraic system of equations with banded coefficient matrix. The desired equation is considered on one and two-dimensional spatial domains. Efficiency, accuracy, and convergence of the proposed method are demonstrated numerically in both cases. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper, the study the momentum and heat transfer characteristics in an incompressible electrically conducting non-Newtonian boundary layer flow of a viscoelastic fluid over a stretching sheet. The partial differential equations governing the flow and heat transfer characteristics are converted into highly nonlinear coupled ordinary differential equations by similarity transformations. The resultant coupled highly nonlinear ordinary differential equations are solved by means of, homotopy analysis method (HAM) for constructing an approximate solution of heat transfer in magnetohydrodynamic (MHD) viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source. The proposed method is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiry parameter, which provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper, we study the initial boundary-value problem with combined power-type nonlinearities by utilizing potential well method. We provide an algorithm to compute the depth of the potential well with the help of Mathematica, and derive the invariant subsets, global existence and blowup of solutions. Moreover, we obtain the invariant subsets, global existence and blowup of solutions for the critical case. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, a new technique of homotopy analysis method (HAM) is proposed for solving high-order nonlinear initial value problems. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM) by transform the nth-order nonlinear differential equation to a system of n first-order equations. Second- and third- order problems are solved as illustration examples of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.

This paper is an advanced extension of the work reported in (Nonlinear Anal. 2005; 63:1467–1473). A transport equation that describes the propagation of a substance in a moving fluid or gas is considered. The equation contains the transient, convection, and diffusion terms. The problem is formulated in a bounded domain provided with an inlet and an outlet for the fluid or gas flow. The crucial point of the problem setting is a hysteresis-type condition posed on an active part of the boundary. This condition reflects the nondecreasing accumulation with saturation of the transported substance at each point of the active boundary part. We prove the existence and uniqueness of solutions to this problem, study the regularity properties of solutions, and perform numerical simulations that clarify the behavior of the model. Comparing with the results of (Nonlinear Anal. 2005; 63:1467–1473), the advancement of this work consists in accounting for the motion of the fluid or gas and posing inlet and outlet boundary conditions. Copyright © 2009 John Wiley & Sons, Ltd.

In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19:1263–1317; Nonlinear Anal. 1997; 28:1299–1322; Chin. Ann. Math. 2004; 25B:37–56). We give a new, very simple proof of this result and also give a sharp point-wise decay estimate of the solution. Then, we consider the mixed initial-boundary-value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12(1):59–78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point-wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd.

In this paper, we study the existence of positive solutions for the following Sturm-Liouville-like four-point singular boundary value problem (BVP) with p-Laplacian (φ p (u'(t)))'+q(t)f(u(t))=0, t∈(0,1) u(0)-αu'(ξ)=0, u(1)+βu'(η)=0 where φ p (s) = |s| p-2 s, p > 1, f is a lower semi-continuous function. Using the fixed-point theorem of cone expansion and compression of norm type, the existence of positive solution and infinitely many positive solutions for Sturm-Liouville- like singular BVP with p-Laplacian are obtained.

In this paper, we derive bounds for the solutions of a quasilinear heat equation in a finite cylindrical region if the far end and the lateral surface are held at zero temperature, and a nonzero temperature is applied at the near end. Some continuous dependence inequalities are also obtained. We also investigate the case in which a given heat flux is prescribed at the near end, instead of a given temperature. Copyright © 2008 John Wiley & Sons, Ltd.

This paper is devoted to the existence and regularity of the homogenous Dirichlet boundary value problem for a singular nonlinear elliptic equation with natural growth in the gradient. By certain transformations, the problem can be transformed formally into either a Dirichlet problem or boundary blowup problems without gradient term, for which the corresponding existence results are also derived, which is a partial extension and supplement to the previous works. Copyright © 2008 John Wiley & Sons, Ltd.

This paper deals with the initial and boundary value problem for a singular nonlinear parabolic equation. The existence of solutions is established by parabolic regularization. Some properties of solutions, for instance localization and large time behaviour are also discussed. Copyright © 2007 John Wiley & Sons, Ltd.

In this paper we are concerned with the existence and energy decay of solution to the initial boundary value problem for the coupled Klein–Gordon–Schrodinger equations with non-linear boundary damping and memory term. Copyright © 2006 John Wiley & Sons, Ltd.

In this paper we establish some results regarding the existence of solution on L1 spaces to a nonlinear boundary value problem originally proposed by Lebowitz and Rubinow (J. Math. Biol. 1974; 1:17–36) to model an age-structured proliferating cell population. Our approach, based on topological methods, uses essentially the specific properties of weakly compact sets on L1 spaces. Our results provide positive answers to the questions posed in Jeribi (Nonlinear Anal. Real World Appl. 2002; 3:85–105). Copyright © 2005 John Wiley & Sons, Ltd.

In this paper, we discuss the existence and uniqueness of bounded weak solution for non-linear parabolic boundary value problem with equivalued surface and correct the mistake in Zhang Xu (Math. Meth. Appl. Sci. 1999; 22: 259). The approach is based on L∞ estimate of solution. Copyright © 2004 John Wiley & Sons, Ltd.

In this paper the degenerate parabolic system ut=u(uxx+av). vt=v(vxx+bu) with Dirichlet boundary condition is studied. For , the global existence and the asymptotic behaviour (α1=α2) of solution are analysed. For , the blow-up time, blow-up rate and blow-up set of blow-up solution are estimated and the asymptotic behaviour of solution near the blow-up time is discussed by using the ‘energy’ method. Copyright © 2003 John Wiley & Sons, Ltd.

The paper studies the existence, asymptotic behaviour and stability of global solutions to the initial boundary value problem for a class of strongly damped non-linear wave equations. By a H00.5ptk-Galerkin approximation scheme, it proves that the above-mentioned problem admits a unique classical solution depending continuously on initial data and decaying to zero as t+∞as long as the non-linear terms are sufficiently smooth; they, as well as their derivatives or partial derivatives, are of polynomial growth order and the initial energy is properly small. Copyright © 2003 John Wiley & Sons, Ltd.

This paper is concerned with the standing wave in coupled non-linear Klein–Gordon equations. By an intricate variational argument we establish the existence of standing wave with the ground state. Then we derive out the sharp criterion for blowing up and global existence by applying the potential well argument and the concavity method. We also show the instability of the standing wave. Copyright © 2003 John Wiley & Sons, Ltd.

In this paper we consider a system of heat equations u t, = xu, v t = xv in an unbounded domain Ω ⊂ R N coupled through the Neumann boundary conditions u v = v P , v v = u q , where p > 0,q > 0, pq > 1 and v is the exterior unit normal on ∂Ω. It is shown that for several types of domain there exists a critical exponent such that all of positive solutions blow up in a finite time in subcritical case (including the critical case) while there exist positive global solutions in the supercritical case if initial data are small.

In this paper we investigate the properties of positive solutions for three non-local reaction-diffusion problems. The conditions on the existence and non-existence of global positive solutions are given. Moreover, we prove that the blow-up set is the whole region when the non-linearity occurs in the equation.

In this paper, we consider non-negative solutions of We prove that if pq ⩽ 1 every solution remains bounded, while all solutions blow up in a finite time if pq > 1. We also show that if p,q > 1, then blow-up can only occur on the boundary.

In this paper we study the motion of slightly compressible inviscid fluids. We prove that the solution of the corresponding system of nonlinear partial differential equations converges (uniformly) in the strong norm (that of the data space) to the solution of the incompressible equations, as the Mach number goes to zero (see Theorem 1.2). Actually, our proof applies to a large class of singular limit problems as shown in the Theorem 2.2.

The paper is devoted to a posteriori quantitative analysis for errors caused by linearization of non-linear elliptic boundary value problems and their finite element realizations. We employ duality theory in convex analysis to derive computable bounds on the difference between the solution of a non-linear problem and the solution of the linearized problem, by using the solution of the linearized problem only. We also derive computable bounds on differences between finite element solutions of the nonlinear problem and finite element solutions of the linearized problem, by using finite element solutions of the linearized problem only. Numerical experiments show that our a posteriori error bounds are efficient.

This paper is concerned with bifurcation from infinity for nonlinear elliptic equations, which are not necessarily linearizable at infinity. The methods employed are global perturbation techniques by means of which one obtains access to continua of positive solutions bifurcating from infinity via continua bifurcating from trivial solutions.

The paper deals with the three-dimensional Dirichlet boundary value problem (BVP) for a second order strongly elliptic self-adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary-domain integral equations (LBDIEs). The equivalence between the Dirichlet BVP and the corresponding LBDIE system is studied. We establish that the obtained localized boundary-domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener-Hopf factorization method we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces., 31 pages, Math. Meth. Appl. Sci., 2016, 21p