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In this paper, we study the growth behaviour of entire Clifford algebra-valued solutions to iterated Dirac and generalized Cauchy–Riemann equations in higher-dimensional Euclidean space. Solutions to this type of systems of partial differential equations are often called k-monogenic functions or, more generically, polymonogenic functions. In the case dealing with the Dirac operator, the function classes of polyharmonic functions are included as particular subcases. These are important for a number of concrete problems in physics and engineering, such as, for example, in the case of the biharmonic equation for elasticity problems of surfaces and for the description of the stream function in the Stokes flow regime with high viscosity. Furthermore, these equations in turn are closely related to the polywave equation, the poly-heat equation and the poly-Klein–Gordon equation. In the first part we develop sharp Cauchy-type estimates for polymonogenic functions, for equations in the sense of Dirac as well as Cauchy–Riemann. Then we introduce generalizations of growth orders, of the maximum term and of the central index in this framework, which in turn then enable us to perform a quantitative asymptotic growth analysis of this function class. As concrete applications we develop some generalizations of some of Valiron's inequalities in this paper. Copyright © 2006 John Wiley & Sons, Ltd.

The p-Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p-Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p-Laplace equation for 1 < p < 2 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the p-Laplace equation into the p-Dirac equation. This equation will be solved iteratively by using a fixed-point theorem. Applying operator-theoretical methods for the p-Dirac equation and p-Laplace equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider the analytical solutions of fractional partial differential equations (PDEs) with Riesz space fractional derivatives on a finite domain. Here we considered two types of fractional PDEs with Riesz space fractional derivatives such as Riesz fractional diffusion equation (RFDE) and Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order α∈(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order β∈(0,1] and of order α∈(1,2] respectively. Here the analytic solutions of both the RFDE and RFADE are derived by using modified homotopy analysis method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. Here the space fractional derivatives are defined as Riesz fractional derivatives. Copyright © 2015 John Wiley & Sons, Ltd.

This paper considers an abstract third-order equation in a Hilbert space that is motivated by, and ultimately directed to, the “concrete” Moore–Gibson–Thompson Equation arising in high-intensity ultrasound. In its simplest form, with certain specific values of the parameters, this third-order abstract equation (with unbounded free dynamical operator) is not well-posed. In general, however, in the present physical model, a suitable change of variable permits one to show that it has a special structural decomposition, with a precise, hyperbolic-dominated driving part. From this, various attractive dynamical properties follow: s.c. group generation; a refined spectral analysis to include a specifically identified point in the continuous spectrum of the generator (so that it does not have compact resolvent) as an accumulation point of eigenvalues; and a consequent theoretically precise exponential decay with the same decay rate in various function spaces. In particular, the latter is explicit and sharp up to a finite number of (stable) eigenvalues of finite multiplicity. A computer-based analysis confirms the theoretical spectral analysis findings. Moreover, it shows that the dynamic behavior of these unaccounted for finite-dimensional eigenvalues are the ones that ultimately may dictate the rate of exponential decay, and which can be estimated with arbitrarily preassigned accuracy. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper, we investigate the existence of multiple solutions to a second-order Dirichlet boundary-value problem with impulsive effects. The proof is based on critical point theorems. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, we mainly study the Rm (m>0) Riemann boundary value problems for functions with values in a Clifford algebra Cl(V3, 3). We prove a generalized Liouville-type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k-monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the Rm (m>0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright © 2009 John Wiley & Sons, Ltd.

In this paper, the method of deriving the Backlund transformation from the Riccati form of inverse method is presented for the perturbed nonlinear Schrodinger equation (PNSE). Consequently, the exact solutions for the PNSE can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions that are solutions of the associated AKNS, that is, a linear eigenvalues problem in the form of a system of partial differential equation. Moreover, we construct a new soliton solution from the old one and its wave function. Copyright © 2008 John Wiley & Sons, Ltd.

In this paper we consider a generalization of the classical time-harmonic Maxwell equations, which as an additional feature includes a radial symmetric perturbation in the form of the Euler operator E := Sigma(i)x(i)partial derivative/partial derivative x(i). We show how one can apply hypercomplex analysis methods to solve the partial differential equation (PDE) system [D-lambda-alpha E]f = 0, where D is the Euclidean Dirac operator and where lambda and alpha are arbitrary non-zero complex numbers, When alpha is a positive real number, the vector-valued solutions to this PDE system provide us precisely with the solutions to the time-harmonic Maxwell equations oil the sphere of radius 1/alpha. We give a fully explicit description of the regular solutions around the origin for general complex alpha, lambda is an element of C\{0} in terms of hypergeometric functions and special homogeneous monogenic polynomials. We also discuss the limit cases lambda -> 0 and alpha -> 0. in the latter one, we actually recognize the classical time-harmonic Solutions to the Maxwell equations in the Euclidean space.

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.

In this paper, we deal with Sturm–Liouville-type problems when the potential of the differential equation may have discontinuity at one inner point and the eigenparameter appears not only in the differential equation, but also in both boundary and transmission conditions. By modifying some techniques of (Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edin. 1977; 77A:293–308; Eigenfunction Expenses Associated with Second-Order Differential Equations I (2nd edn). Oxford University Press: London, 1962) we generalize some results of the classic regular Sturm–Liouville problems. In particular, we construct Green's function, and derive asymptotic approximation formulae for Green's function. Further, we introduce a new operator-theoretic formulation in suitable Hilbert space such a way that the considered problem can be interpreted as the eigenvalue problem of this operator and constract the resolvent of this operator in terms of Green's function. Finally, we estimate the norm of resolvent of this operator. Copyright © 2007 John Wiley & Sons, Ltd.

We consider a model for one-dimensional transversal oscillations of an elastic-ideally plastic beam. It is based on the von Mises model of plasticity and leads after a dimensional reduction to a fourth-order partial difierential equation with a hysteresis operator of Prandtl-Ishlinskii type whose weight function is given explicitly. In this paper, we study the case of clamped beams involving a kinematic hardening in the stress-strain relation. As main result, we prove the existence and uniqueness of a weak solution. The method of proof, based on spatially semidiscrete approximations, strongly relies on energy dissipation properties of one-dimensional hysteresis operators.

In this paper, we consider some Lorenz-gauged vector potential formulations of the eddy-current problem for the time-harmonic Maxwell equations with material properties having only L?-regularity.We prove that there exists a unique solution of these problems, and we show the convergence of a suitable finite element approximation scheme. Moreover, we show that some previously proposed Lorenz-gauged formulations are indeed formulations in terms of the modified magnetic vector potential, for which the electric scalar potential is vanishing.

This paper studies the scattering of electromagnetic waves from a (local) perturbation of a fixed surface, the boundary of a given obstacle in ℝ3. The goal is to produce an algorithm for solving boundary value problems in the exterior of the perturbed domain solely based on the knowledge of the Green function for the original surface. This is done by solving a boundary integral equation which only involves the perturbed portion of the boundary. Copyright © 2006 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 further results on the asymptotic growth behaviour of entire solutions to iterated Dirac equations in ℝn. Solutions to this type of systems of partial differential equations are often called polymonogenic or k-monogenic. In the particular cases where k is even, one deals with polyharmonic functions. These are of central importance for a number of concrete problems arising in engineering and physics, such as for example in the case of the biharmonic equation for the description of the stream function in the Stokes flow regime with low Reynolds numbers and for elasticity problems in plates. The asymptotic study that we are going to perform within the context of these PDE departs from the Taylor series representation of their solutions. Generalizations of the maximum term and the central index serve as basic tools in our analysis. By applying these tools we then establish explicit asymptotic relations between the growth behaviour of polymonogenic functions, the growth behaviour of their iterated radial derivatives and that of functions obtained by applying iterations of the Γ operator to them. Copyright © 2005 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.

In this paper, the unique solvability, Fredholm property, and the principle of limiting absorption are proved for a boundary value problem for the system of Maxwell's equations in a semi-infinite rectangular cylinder coupled with a layer by an aperture of arbitrary shape. Conditions at infinity are taken in the form of the Sveshnikov–Werner partial radiation conditions. The method of solution employs Green's functions of the partial domains and reduction to vector pseudodifferential equations considered in appropriate vectorial Sobolev spaces. Singularities of Green's functions are separated both in the domain and on its boundary. The smoothness of solutions is established. 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.

The objective of this paper aims to prove positivity of solutions for a semilinear dissipative partial differential equation with non-linear diffusion. The equation is a generalized model of the well-known Fisher–Kolmogorov equation and represents a class of dissipative partial differential equations containing differential operators of higher order than the Laplacian. It arises in a variety of meaningful physical situations including gas flows, diffusion of an electron–ion plasma and the dynamics of biological populations whose mobility is density dependent. In all these situations, the solutions of the equation must be positive functions. Copyright © 2002 John Wiley & Sons, Ltd.

In this paper the authors establish continuous dependence of the temperature on the spatial geometry in an initial-boundary value problem for the generalized Maxwell–Cattaneo system of equations. Copyright © 2001 John Wiley & Sons, Ltd.

Continuous dependence on a modelling parameter is established for solutions of a problem for a Ginzburg–Landau equation. Both the forward and backward in time problems are analysed. In the former case, we derive a priori estimates that indicate that solutions depend continuously on a parameter in the governing differential equation. For the backward in time problem, which is ill-posed, we obtain Holder continuous dependence on the parameter provided we assume that the solutions exist and are properly constrained. In the last part of the paper, nonexistence results for a backward in time problem for the Ginzburg–Landau equation are discussed. Copyright © 2000 John Wiley & Sons, Ltd.

This paper deals with the regularity of the global attractor fbr the Klein-Gordon-Schrodinger equation. Using a decomposition method, we prove that the global attractor for the one-dimensional model consists of smooth functions provided the forcing terms are regular.

The existence of global weak solutions to the degenerate problem describing Joule's heating in a current and heat conductive medium is proved via the Galerkin method. The existence proof proceeds by a sequence of a priori estimates, which may be achieved by the standard method. Under suitable hypotheses on the electrical conductivity the boundedness in L r (Q T ) of the absolute temperature of the medium is established by the method of Stampacchia, The paper is an extension of the work by Cimatti [3, 4], Shi et al. [12] and Xu [16]. This extension consists of employing the equations for the electric field E, derived from Maxwell's equations, instead of the equation for the electric potential, with the appropriate modification in the energy balance equation.

In this paper, constitutive equations of non-local type are coupled with Maxwell equations and the resulting differential problem is studied. A weak formulation is given for an initial-boundary-value problem for Maxwell equations in a medium obeying such constitutive equations with perfectly conducting boundary, and it is shown that such a problem admits at most one solution. The uniqueness theoren is then shown to imply the density of the range of a certain operator in the space of solutions and this result, together with an a priori energy inequality, is used to prove existence of solutions. Then the study of asymptotic stability of solutions is addressed. In particular, solutions are shown to be L 2 in time over (0, ∞). Finally, a brief description is given of the alternative problem arising when more general constitutive equations are used.

We study the slow steady-state flow of a fluid of Prandtl-Eyring type and prove (partial) regularity of the strain velocity by investigating an appropriate variational problem. We further discuss local minimizers of variational integrals which occur in the theory of plasticity with logarithmic hardening. For this model we show that the deformation gradient in the three-dimensional case is smooth up to a closed set of vanishing Lebesgue measure. The paper also presents an introduction into various function spaces which are needed to formulate the problems. Copyright © 1999 John Wiley & Sons, Ltd.

We obtain in this paper the global boundedness of solutions to a Fujita-type reaction-diffusion system. This global boundedness results from diffusion effect, homogeneous Dirichlet boundary value conditions and appropriate reactions.

This paper is concerned with the solution of Maxwell equations in the modelling of the scattering of a time-harmonic electromagnetic wave by an obstacle located in a two-layered medium. The use of the Silver-Muller radiation condition in each layer is shown to provide a well-posed scattering problem. The analysis is based on the study of the Green tensor, which allows to relate the radiation condition to an integral representation formula. The analyticity properties of the scattering problem with respect to the frequency are then investigated. This gives rise to a limiting absorption principle and furnishes a characterization of the resonances.

The paper deals with the Fourier-finite-element method (FFEM), which combines the approximate Fourier method with the finite-element method, and its application to Poisson-like equations -pΔ 3 u = f in three-dimensional axisymmetric domains Ω. Here, p is a piecewise constant coefficient having a jump at some axisymmetric interface. Special emphasis is given to estimates of the Fourier-finite-element error in the Sobolev space H 1 (Ω), if the interface is smooth or if it meets the boundary of Ω at some edge. In general, the solution u contains a singularity at the interface, which is described by a tensor product representation and treated numerically by appropriate mesh grading in the meridian plane of Ω. The rate of convergence of the combined approximation in H 1 (Ω) is proved to be O(h + N -1 ) (h, N : the parameters of the finite-element- and Fourier-approximation, with h →0, N →∞). The theoretical results are confirmed by numerical experiments.

In this paper the global existence of weak solutions for the Vlasov-Poisson-Fokker-Planck equations in three dimensions is proved with an L 1 ∩ L p initial data. Also, the global existence of weak solutions in four dimensions with small initial data is studied. A convergence of the solutions is obtained to those built by E. Horst and R. Hunze when the Fokker-Planck term vanishes. In order to obtain the a priori necessary estimates a sequence of approximate problems is introduced. This sequence is obtained starting from a non-linear regularization of the problem together with a linearization via a time retarded mollification of the non-linear term. The a priori bounds are reached by means of the control of the kinetic energy in the approximate sequence of problems. Then, the proof is completed obtaining the equicontinuity properties which allow to pass to the limit.

In this paper, a tracking method is proposed for the expansion of gas flow into vacuum which may be combined with numerical methods for the equations of gas dynamics, the Euler equations. This tracking prevents the difficulties of the numerical approximation introduced by the vacuum as a region where the Euler equations are not valid due to the failure of the continuum assumption. The tracking algorithm is based on the exact or an approximate solution of the vacuum Riemann problem. This is the initial value problem with two constant states, one being the gas and the other the vacuum state, and a limit case of the usual Riemann problem. In this approach, the gas–vacuum boundary is sharply resolved within one mesh interval. For a test problem, the numerical results of gas flow into vacuum are presented which indicate that the gas vacuum boundary is captured very well.

In this paper, we investigate Poincare type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. We find exact and easily computable constants in these inequalities for some basic domains (rectangles, cubes, and right triangles) and discuss applications of the inequalities to quantitative analysis of partial differential equations. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper we study a coupled non-linear system of partial differential equations that models the dynamics of structural phase transitions in a one-dimensional non-viscous and heat-conducting solid. The corresponding Helmholtz free energy density is assumed in Ginzburg–Landau form; to allow for phase transitions and hysteresis phenomena, it is not assumed convex in the order parameter. It is shown that the solution of the system depends continuously upon the data, and we prove an existence result for an associated optimal control problem.