Showing papers on "Integro-differential equation published in 2007"
Book•
15 Jul 2007TL;DR: Important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral equation research.
Abstract: Integral Equation Methods for Electromagnetic and Elastic Waves is an outgrowth of several years of work. There have been no recent books on integral equation methods. There are books written on integral equations, but either they have been around for a while, or they were written by mathematicians. Much of the knowledge in integral equation methods still resides in journal papers. With this book, important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral equation research. Also, learning the fundamentals of linear elastic wave theory does not require a quantum leap for electromagnetic practitioners. Integral equation methods have been around for several decades, and their introduction to electromagnetics has been due to the seminal works of Richmond and Harrington in the 1960s. There was a surge in the interest in this topic in the 1980s (notably the work of Wilton and his coworkers) due to increased computing power. The interest in this area was on the wane when it was demonstrated that differential equation methods, with their sparse matrices, can solve many problems more efficiently than integral equation methods. Recently, due to the advent of fast algorithms, there has been a revival in integral equation methods in electromagnetics. Much of our work in recent years has been in fast algorithms for integral equations, which prompted our interest in integral equation methods. While previously, only tens of thousands of unknowns could be solved by integral equation methods, now, tens of millions of unknowns can be solved with fast algorithms. This has prompted new enthusiasm in integral equation methods.
473 citations
244 citations
TL;DR: In this paper, the variational iteration method is applied to solve integro-differential equations, and the results show that the method provides a straightforward and powerful mathematical tool for solving various integro differential equations.
175 citations
TL;DR: The non-local peridynamic theory as mentioned in this paper describes the displacement field of a continuous body by the initial value problem for an integro-differential equation that does not include any spatial derivative.
Abstract: The non-local peridynamic theory describes the displacement field of a continuous body by the initial-value problem for an integro-differential equation that does not include any spatial derivative. The non-locality is determined by the so-called peridynamic horizon $\delta$ which is the radius of interaction between material points taken into account. Well-posedness and structural properties of the peridynamic equation of motion are established for the linear case corresponding to small relative displacements. Moreover the limit behavior as $\delta \rightarrow 0$ is studied.
175 citations
TL;DR: This article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson-Boltzmann equation with delta distribution sources, and it introduces the first provably convergent adaptive method for the equation.
Abstract: A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson-Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson-Boltzmann equation is introduced as an auxiliary problem, making it possible to study the original nonlinear equation with delta distribution sources. A priori error estimates for the finite element approximation are obtained for the regularized Poisson-Boltzmann equation based on certain quasi-uniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge. The Poisson-Boltzmann equation does not appear to have been previously studied in detail theoretically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson-Boltzmann equation. Note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson-Boltzmann equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems.
162 citations
TL;DR: In this article, exact solutions for some nonlinear evolution equations are obtained based on the Exp-function method, including the KdV equation, the Burgers' equation, and the combined kdV-mKdV equations.
157 citations
TL;DR: In this paper, the generalized fractional Schrodinger equation with space and time fractional derivatives is constructed and solved for free particle and for a square potential well by the method of integral transforms, Fourier transform and Laplace transform, and the solution can be expressed in terms of Mittag-Leffler function.
Abstract: In this paper the generalized fractional Schrodinger equation with space and time fractional derivatives is constructed. The equation is solved for free particle and for a square potential well by the method of integral transforms, Fourier transform and Laplace transform, and the solution can be expressed in terms of Mittag-Leffler function. The Green function for free particle is also presented in this paper. Finally, we discuss the relationship between the cases of the generalized fractional Schrodinger equation and the ones in standard quantum.
147 citations
TL;DR: In this article, a generalized impedance boundary condition is developed to rigorously model on-chip interconnects in the full-wave surface integral equation by a two-region formulation.
Abstract: A generalized impedance boundary condition is developed to rigorously model on-chip interconnects in the full-wave surface integral equation by a two-region formulation. It is a combination of the electric-field integral equation for the exterior region and the magnetic-field integral equation for the interior conductive region. The skin effect is, therefore, well captured. A novel integration technique is proposed to evaluate the Green's function integrals in the conductive medium. Towards tackling large-scale problems, the mixed-form fast multipole algorithm and the multifrontal method are incorporated. A new scheme of the loop-tree decomposition is also used to alleviate the low-frequency breakdown for the formulation. Numerical examples show the accuracy and reduced computation cost.
142 citations
TL;DR: In this article, Topaz et al. developed a satisfactory existence theory for a general class of aggregation equations, which is a non-linear, non-local partial differential equation that is a regularization of a backward diffusion process.
Abstract: The purpose of this work is to develop a satisfactory existence theory for a general class of aggregation equations. An aggregation equation is a non-linear, non-local partial differential equation that is a regularization of a backward diffusion process. The non-locality arises via convolution with a potential. Depending on how regular the potential is, we prove either local or global existence for the solutions. Aggregation equations have been used recently to model the dynamics of populations in which the individuals attract each other (Bodnar and Velazquez, 2005; Holm and Putkaradze, 2005; Mogilner and Edelstein-Keshet, 1999; Morale et al., 2005; Topaz and Bertozzi, 2004; Topaz et al., 2006).
124 citations
TL;DR: In this article, an initial-value problem for an integro-differential equation (IDE) that incorporates non-local effects is formulated and well-posedness in L ∞(ℝ) as well as jump relations is proved.
Abstract: Long-range interactions for linearly elastic media resulting in nonlinear dispersion relations are modeled by an initial-value problem for an integro-differential equation (IDE) that incorporates non-local effects. Interpreting this IDE as an evolutionary equation of second order, well-posedness in L ∞(ℝ) as well as jump relations are proved. Moreover, the construction of the micromodulus function from the dispersion relation is studied. A numerical approximation based upon quadrature is suggested and carried out for two examples, one involving jump discontinuities in the initial data corresponding to a Riemann-like problem.
108 citations
TL;DR: In this article, different spatial discretisation methods for solving the peridynamic equation of motion are suggested for a linear microelastic material of infinite length in one spatial dimension and the conservation of energy is studied for the continuous as well as discretised problem.
Abstract: Different spatial discretisation methods for solving the peridynamic equation of motion are suggested. The methods proposed are tested for a linear microelastic material of infinite length in one spatial dimension. Moreover, the conservation of energy is studied for the continuous as well as discretised problem.
TL;DR: In this article, the CAS wavelet operational matrix P of integration is presented and a general procedure to generate this matrix P is given, which is then utilized to reduce the integro-differential equation to the algebraic equations.
TL;DR: In this article, a diffusion wave equation with two fractional derivatives of different order on bounded and unbounded spatial domains is analyzed and solutions to signalling and Cauchy problems in terms of a series and integral representation are given.
Abstract: We analyse a diffusion wave equation with two fractional derivatives of different order on bounded and unbounded spatial domains. Thus, our model represents a generalized telegraph equation. Solutions to signalling and Cauchy problems in terms of a series and integral representation are given. Classical wave and heat conduction equations are obtained as limiting cases.
TL;DR: In this article, numerical methods for solving integro-differential equations which generalize the well-known Fisher equation were studied. And the stability and convergence of the methods are studied.
TL;DR: In this article, a rigorous dual-porosity model is formulated, which accurately represents the coupling between large-scale fractures and the micropores within dual porosity media, and the combined flow equation is a nonlinear integro-(partial) differential equation.
Abstract: In this paper a rigorous dual-porosity model is formulated, which accurately represents the coupling between large-scale fractures and the micropores within dual porosity media. The overall structure of the porous medium is conceptualized as being blocks of diffusion dominated micropores separated by natural fractures (e.g. cleats for coal) through which Darcy’s flow occurs. In the developed model, diffusion in the matrix blocks is fully coupled to the pressure distribution within the fracture system. Specific assumptions on the pressure behaviour at the matrix boundary, such as step-time function employed in some earlier studies, are not invoked. The model involves introducing an analytical solution for diffusion within a matrix block, and the resultant combined flow equation is a nonlinear integro-(partial) differential equation. Analyses to the equation in this text, in addition to the theoretical development of the proposed model, include: (1) discussion on the “fading memory” of the model; (2); one-dimensional perturbation solution subject to a specific condition; and (3) asymptotic analyses of the “long-time” and “short-time” responses of the flow. Two previous models, the Warren-Root and the modified Vermeulen models, are compared with the proposed model. The advantages of the new model are demonstrated, particularly for early time prediction where the approximations of these other models can lead to significant error.
TL;DR: In this article, a second-order radiative transfer equation is derived, which is a diffusion-type equation similar to the heat conduction equation for an anisotropic medium.
Abstract: The original radiative transfer equation is a first-order integrodifferential equation, which can be taken as a convection-dominated equation. The presence of the convection term may cause nonphysical oscillation of solutions. This type of instability can occur in many numerical methods, including the finite-difference method and the finite-element method, if no special stability treatment is used. To overcome this problem, a second-order radiative transfer equation is derived, which is a diffusion-type equation similar to the heat conduction equation for an anisotropic medium. The consistency of the second-order radiative transfer equation with the original radiative transfer equation is demonstrated. The perturbation characteristics of error are analyzed and compared for both the first- and second-order equations. Good numerical properties are found for the second-order radiative transfer equation. To show the properties of the numerical solution, the standard Galerkin finite-element method is employed ...
TL;DR: In this article, the second-order nonlinear equations of Rabelo which describe pseudospherical surfaces were studied and their general solutions were obtained by transforming these equations to the constant-characteristic form.
Abstract: We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The other two are transformed into a linear equation and the Liouville equation, and in this way their general solutions are obtained.
TL;DR: A new method to solve the linear integro-differential equations using radial basis function (RBF) networks by means of RBFs, whose coefficients are computed by training a RBF network, and is written as a sum of two parts.
Journal Article•
TL;DR: In this paper, the fractional order differential equation with Riemann-Liouville fractional derivatives is treated as the Euler-Lagrange equation in variational principles.
Abstract: We treat the fractional order differential equation that contains the left and right Riemann-Liouville fractional derivatives. Such equations arise as the Euler-Lagrange equation in variational principles with fractional derivatives. We reduce the problem to a Fredholm integral equation and construct a solution in the space of continuous functions. Two competing approaches in formulating differential equations of fractional order in Mechanics and Physics are compared in a specific example. It is concluded that only the physical interpretation of the problem can give a clue which approach should be taken. Mathematics Subject Classification: 26A33; 70H03, 70H25, 70S05; 49S05
TL;DR: In this article, a simple polynomial approximation of a function is employed to obtain approximate solution of a class of singular integral equations of the second kind, and convergence of the method for both the equations is also established.
TL;DR: In this article, a meshless regularized integral equation method (MRIEM) was developed to solve the interior and exterior Dirichlet problems for the two-dimensional Laplace equation, which consists of three parts: Fourier series expansion, second kind Fredholm integral equation and an analytically regularized solution of the unknown boundary condition on an artificial circle.
Abstract: Summary A new method is developed to solve the interior and exterior Dirichlet problems for the two-dimensional Laplace equation, namely the meshless regularized integral equation method (MRIEM), which consists of three parts: Fourier series expansion, the second kind Fredholm integral equation and an analytically regularized solution of the unknown boundary condition on an artificial circle. We find that the new method is powerful even for the problem with very complex boundary shape and with boundary noise.
TL;DR: In this article, the direct scattering problem for time harmonic electromagnetic fields in a layered medium where arbitrary incident fields are scattered by a medium described by a space dependent permittivity and conductivity was studied.
Abstract: In the first part of this paper we study the direct scattering problem for time harmonic electromagnetic fields in a layered medium where arbitrary incident fields are scattered by a medium described by a space dependent permittivity and conductivity. We derive an integral equation and prove a theorem of Riesz-Fredholm type. In the second part we investigate the Factorization Method for the corresponding inverse problem with magnetic dipoles as incident fields. This is the problem to recover the support of the contrast from field measurements.
TL;DR: In this paper, the authors consider a given homogeneous equation and a corresponding equation with forcing term and give new conditions implying that the latter equation inherits the oscillatory behavior of the homogenous equation.
Abstract: Oscillation and nonoscillation properties of second order Sturm?Liouville dynamic equations on time scales ? for example, second order self-adjoint differential equations and second order Sturm?Liouville difference equations ? have attracted much interest. Here we consider a given homogeneous equation and a corresponding equation with forcing term. We give new conditions implying that the latter equation inherits the oscillatory behavior of the homogeneous equation. We also give new conditions that introduce oscillation of the inhomogeneous equation while the homogeneous equation is nonoscillatory. Finally, we explain a gap in a result given in the literature for the continuous and the discrete case. A more useful result is presented, improving the theory even for the corresponding continuous and discrete cases. Examples illustrating the theoretical results are supplied.
01 Jun 2007
TL;DR: In this article, the authors studied the maximal regularity of an integro-differential equation with infinite delay on periodic vector-valued Besov spaces and used Fourier multipliers to characterize periodic solutions solely in terms of spectral properties on the data.
Abstract: Maximal regularity for an integro-differential equation with infinite delay on periodic vector-valued Besov spaces is studied. We use Fourier multipliers techniques to characterize periodic solutions solely in terms of spectral properties on the data. We study a resonance case obtaining a compatibility condition which is necessary and sufficient for the existence of periodic solutions.
TL;DR: The work proposes and studies a model for one-dimensional spatially extended systems, which involve nonlocal interactions and finite propagation speed, and shows that the general reaction-diffusion equation, the Swift-Hohenberg equation, and the general Kuramoto-Sivashinsky equation represent special cases of the proposed model for limited spatial interaction ranges and for infinite propagation speeds.
Abstract: The work proposes and studies a model for one-dimensional spatially extended systems, which involve nonlocal interactions and finite propagation speed. It shows that the general reaction-diffusion equation, the Swift-Hohenberg equation, and the general Kuramoto-Sivashinsky equation represent special cases of the proposed model for limited spatial interaction ranges and for infinite propagation speeds. Moreover, the Swift-Hohenberg equation is derived from a general energy functional. After a detailed validity study on the generalization conditions, the three equations are extended to involve finite propagation speeds. Moreover, linear stability studies of the extended equations reveal critical propagation speeds and unusual types of instabilities in all three equations. In addition, an extended diffusion equation is derived and studied in some detail with respect to finite propagation speeds. The extended model allows for the explanation of recent experimental results on non-Fourier heat conduction in nonhomogeneous material.
TL;DR: A numerical procedure for solving a class of linear integro-differential equation of Fredholm type, using the globally defined Sinc basis functions, and properties of the Sinc procedure are utilized to reduce the computation of the linear integros to some algebraic equations.
TL;DR: In this article, a meshless regularized integral equation method (MRIEM) was developed to solve the Dirichlet problems for the two-dimensional Laplace equation in the doubly-connected domains.
Abstract: A new method is developed to solve the Dirichlet problems for the two-dimensional Laplace equation in the doubly-connected domains, namely the meshless regularized integral equations method (MRIEM), which consists of three portions: Fourier series expansion, the Fredholm integral equations, and linear equations to determine the unknown boundary conditions on artificial circles. The boundary integral equations on artificial circles are singular-free and the kernels are degenerate. When boundary-type methods are inefficient to treat the problems with complicated domains, the new method can be applicable for such problems. The new method by using the Fourier series and the Fourier coefficients can be adopted easily to derive the meshless numerical method. Several numerical examples are tested showing that the new method is powerful. Keyword: Laplace equation, Meshless method, Regularized integral equation, Artificial circles, Doubly-connected domain, Degenerate kernel
TL;DR: In this article, the authors consider the problem of performing the preliminary symmetry classification of a class of quasi-linear PDEs containing one or more arbitrary functions, and provide an easy condition involving these functions in order that nontrivial Lie point symmetries be admitted, and a geometrical characterization of the relevant system of equations determining these symmeterries.
Abstract: We consider the problem of performing the preliminary “symmetry classification” of a class of quasi-linear PDE’s containing one or more arbitrary functions: we provide an easy condition involving these functions in order that nontrivial Lie point symmetries be admitted, and a “geometrical” characterization of the relevant system of equations determining these symmetries. Two detailed examples will elucidate the idea and the procedure: the first one concerns a nonlinear Laplace-type equation, the second a generalization of an equation (the Grad–Schluter–Shafranov equation) which is used in magnetohydrodynamics.
01 Jan 2007
TL;DR: In this paper, the deFinetti-Kolmogorov law is used to define the Ito non-anticipating integral representation and the closed anticipating extension of it.
Abstract: 1. Stochastic measures and functions. Space-time products. Stochastic measures with independent values. The events generated. The deFinetti-Kolmogorov law. Non-anticipating and predictable stochastic functions. 2. The Ito non-anticipating integral. A general definition and related properties. The stochastic Poisson integral. The jumping stochastic processes. Gaussian-Poisson stochastic measures. 3. The non-anticipating integral representation. Multilinear polynomials and Ito multiple integrals. Integral representations with Gaussian-Poisson integrators. Homogeneous integrators. 4. The non-anticipating derivative. A general definition and related properties. Differentiation formulae. 5. Anticipating derivative and integral. Definitions and related properties. The closed anticipating extension of the Ito nonanticipating integral.