Showing papers on "Integro-differential equation published in 2013"
Book•
15 Jul 2013
TL;DR: The Cocycle Equation Appendices: Basic Terminology and Results Substitutes for Commutativity The Casorati Determinant Regularity Matrix-Coefficients of Representations The Small Dimension Lemma Group Cohomology as mentioned in this paper.
Abstract: Contents: Introduction The Additive Cauchy Equation The Multiplicative Cauchy Equation Addition and Subtraction Formulas Levi - Civita's Functional Equation The Symmetrized Sine Addition Formula Equations with Symmetric Right Hand Side The Pre-d'Alembert Functional Equation d'Alembert's Functional Equation d'Alembert's Long Functional Equation Wilson's Functional Equation Jensen's Functional Equation The Quadratic Functional Equation K-Spherical Functions The Sine Functional Equation The Cocycle Equation Appendices: Basic Terminology and Results Substitutes for Commutativity The Casorati Determinant Regularity Matrix-Coefficients of Representations The Small Dimension Lemma Group Cohomology.
157 citations
TL;DR: By use of the extended trial equation method, the exact traveling wave solutions including soliton solution, singular soliton solutions, rational function solution and elliptic integral function solution to one-dimensional general improved KdV (GIKdV) equation and R(m,n) equation are obtained.
102 citations
TL;DR: The results are obtained by using the fixed point technique and solution operator on a complex Banach space and showing the existence, uniqueness and continuous dependence of mild solution for an impulsive neutral fractional order differential equation with infinite delay.
87 citations
TL;DR: In this article, the meshless local radial point interpolation (MLRPI) method is applied to simulate a nonlinear partial integro-differential equation arising in population dynamics.
Abstract: In this paper the meshless local radial point interpolation (MLRPI) method is applied to simulate a nonlinear partial integro-differential equation arising in population dynamics. This PDE 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. In MLRPI method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. The point interpolation method is proposed to construct shape functions using the radial basis functions. A one-step time discretization method is employed to approximate the time derivative. To treat the nonlinearity, a simple predictor–corrector scheme is performed. Also the integral term, which is a kind of convolution, is treated by the cubic spline interpolation. The numerical studies on sensitivity analysis and convergence analysis show that our approach is stable. Finally, two numerical examples are presented showing the behavior of the solution and the efficiency of the proposed method.
83 citations
TL;DR: In this paper, the authors apply the integration-by-parts strategy to all planar three-loop four-point massless on-shell master integrals and show that the differential equations are of the Knizhnik-Zamolodchikov type.
Abstract: We apply a recently suggested new strategy to solve differential equations for master integrals for families of Feynman integrals. After a set of master integrals has been found using the integration-by-parts method, the crucial point of this strategy is to introduce a new basis where all master integrals are pure functions of uniform transcendentality. In this paper, we apply this method to all planar three-loop four-point massless on-shell master integrals. We explicitly find such a basis, and show that the differential equations are of the Knizhnik-Zamolodchikov type. We explain how to solve the latter to all orders in the dimensional regularization parameter epsilon, including all boundary constants, in a purely algebraic way. The solution is expressed in terms of harmonic polylogarithms. We explicitly write out the Laurent expansion in epsilon for all master integrals up to weight six.
75 citations
01 Jan 2013
TL;DR: Feller's paper as discussed by the authors is a rigorous treatment of renewal theory, and to assist the reader his principal results are summarized below in demographic form and notation, and they can be found in Table 1.
Abstract: Feller’s paper is a rigorous treatment of renewal theory, and to assist the reader his principal results are summarized below in demographic form and notation.
74 citations
TL;DR: In this paper, the generalized Schrodinger equation was solved numerically via the Crank-Nicholson scheme, and the stability and convergence of the space fractional variable-order Schroffinger equation were presented in detail.
Abstract: The space fractional Schrodinger equation was further extended to the concept of space fractional variable-order derivative. The generalized equation is very difficult to handle analytically. We solved the generalized equation numerically via the Crank-Nicholson scheme. The stability and the convergence of the space fractional variable-order Schrodinger equation were presented in detail.
57 citations
TL;DR: Comparisons of present results with the previous ones show that the present scheme is more stable and efficient for numerically solving the fourth order partial integro-differential equation with a weakly singular kernel.
48 citations
Journal Article•
TL;DR: In this paper, the existence of mild solutions for the nonlocal Cauchy problem for a class of abstract fractional neutral integro-differential equations with infinite delay is studied. The results are obtained by using the theory of resolvent operators.
Abstract: In this article, we study the existence of mild solutions for the nonlocal Cauchy problem for a class of abstract fractional neutral integro-differential equations with infinite delay The results are obtained by using the theory of resolvent operators Finally, an application is given to illustrate the theory
41 citations
TL;DR: The generalized Hyers–Ulam stability of the Laplace’s equation, △ u = 0 , is proved.
40 citations
TL;DR: It is shown that the Benjamin–Bona–Mahoney (BBM) equation with power law nonlinearity can be transformed by a point transformation to the combined KdV–mKdV equation, that is also known as the Gardner equation.
DOI•
01 Jan 2013
TL;DR: In this paper, the double Laplace transform of the partial fractional integrals and derivatives is used to solve partial differential equations with Caputo fractional derivatives, and the fractional heat equation is investigated in details with the help of this new generalized transform.
Abstract: The heat equation and its fractional generalization are used in various applications in science and engineering. In this paper firstly we introduce the double Laplace transform of the partial fractional integrals and derivatives which can be used to solve partial differential equations with Caputo fractional derivatives. Secondly, the fractional heat equation was investigated in details with the help of this new generalized transform
01 Jan 2013
TL;DR: In this paper, the authors considered the semilinear stochastic heat equation perturbed by additive noise and provided mean square estimates for the overall error, using a non-adaptive wavelet-Galerkin method.
Abstract: We consider the semilinear stochastic heat equation perturbed by additive noise. After time-discretization by Euler’s method the equation is split into a linear stochastic equation and a non-linear random evolution equation. The linear stochastic equation is discretized in space by a non-adaptive wavelet-Galerkin method. This equation is solved first and its solution is substituted into the nonlinear random evolution equation, which is solved by an adaptive wavelet method. We provide mean square estimates for the overall error.
TL;DR: In this paper, a boundary-domain integral diffusion-convection equation has been developed for problems of spatially variable velocity field and spatial variable coefficient, which does not require a calculation of the gradient of the unknown field function, which gives it an advantage over the other known approaches, which are needed and need to be calculated by means of numerical differentiation.
Abstract: In this paper a boundary-domain integral diffusion–convection equation has been developed for problems of spatially variable velocity field and spatially variable coefficient. The developed equation does not require a calculation of the gradient of the unknown field function, which gives it an advantage over the other known approaches, where the gradient of the unknown field function is needed and needs to be calculated by means of numerical differentiation. The proposed equation has been discretized by two approaches—a standard boundary element method, which features fully populated system matrix and matrices of integrals and a domain decomposition approach, which yields sparse matrices. Both approaches have been tested on several numerical examples, proving the validity of the proposed integral equation and showing good grid convergence properties. Comparison of both approaches shows similar solution accuracy. Due to nature of sparse matrices, CPU time and storage requirements of the domain decomposition are smaller than those of the standard BEM approach.
TL;DR: It is shown that the dissipative Kuramoto-Sivashinsky equation does not pass the Painleve test and as consequence this equation is not integrable.
TL;DR: In this paper, a generalized Taylor matrix method is introduced to obtain the approximate solutions of relaxation-oscillation equation by developing the Taylor matrices method, which is based on first taking the truncated fractional Taylor expansions of the functions in the relaxation oscillator and then substituting their matrix forms into the equation.
TL;DR: In this article, a generalization of the 4+2 Davey-Stewartson type system to a 3+1 version, referred to as the 3 + 1 Davey Stewartson equation, is presented.
Abstract: An integrable generalisation of a Davey-Stewartson type system of two equations involving two scalar functions in 4+2, i.e., in four spatial and two temporal dimensions, has been recently derived by one of the authors. Here, we first show that there exists a reduction of this system to a single equation involving a scalar function in 4+2; we will refer to this equation as the 4+2 Davey-Stewartson equation. We then show that it is possible to reduce this equation to an equation in 3+1, which we will refer to as the 3+1 Davey-Stewartson equation. Furthermore, by employing the so-called direct linearising method, we compute 1- and 2-soliton solutions for both the 4+2 and the 3+1 Davey-Stewartson equations.
TL;DR: In this paper, the pseudo-spectral Legendre-Galerkin method (PS-LGM) is applied to solve a nonlinear partial integro-differential equation arising in population dynamics.
Abstract: 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.
TL;DR: In this article, the existence of stationary solutions proof is proved by fixed point and measure theory arguments, and the main result is gained by fixed-point and measure-based arguments.
Abstract: The thermostatted kinetic framework has been recently proposed in [C. Bianca, Nonlinear Analysis: Real World Applications 13 (2012) 2593-2608] for the modeling of complex systems in the applied sciences under the action of an external force field that moves out of equilibrium the system. The framework consists in an integro-differential equation with quadratic nonlinearity coupled with the Gaussian isokinetic thermostat. This paper is concerned with the existence of stationary solutions proof. The main result is gained by fixed point and measure theory arguments. Copyright © 2013 John Wiley & Sons, Ltd.
TL;DR: In this paper, a generalized B-type Kadomtsev-Petviashvili (BKP) equation was proposed, which describes weakly dispersive and small amplitude waves propagating in quasi-two-dimensional media.
Abstract: In this paper we study a -dimensional generalized B-type Kadomtsev-Petviashvili (BKP) equation. This equation is an extension of the well-known Kadomtsev-Petviashvili equation, which describes weakly dispersive and small amplitude waves propagating in quasi-two-dimensional media. We first obtain exact solutions of the BKP equation using the multiple-exp function and simplest equation methods. Furthermore, the conservation laws for the BKP equation are constructed by using the multiplier method.
TL;DR: The solution of the first order nonlinear Fredholm–Volterra integro-differential equation is approximate by means of a new method for its numerical resolution, using the properties of a biorthogonal system in a Banach space and the Banach fixed point theorem.
TL;DR: In this paper, a new integro-differential equation is constructed to describe the Rossby solitary waves in deep rotational fluids, and the effect of detuning parameter and dissipation on Rossby single wave generated by topography is discussed.
Abstract: From rotational potential vorticity-conserved equation with topography effect and dissipation effect, with the help of the multiple-scale method, a new integro-differential equation is constructed to describe the Rossby solitary waves in deep rotational fluids. By analyzing the equation, some conservation laws associated with Rossby solitary waves are derived. Finally, by seeking the numerical solutions of the equation with the pseudospectral method, by virtue of waterfall plots, the effect of detuning parameter and dissipation on Rossby solitary waves generated by topography are discussed, and the equation is compared with KdV equation and BO equation. The results show that the detuning parameter.. plays an important role for the evolution features of solitary waves generated by topography, especially in the resonant case; alpha large amplitude nonstationary disturbance is generated in the forcing region. This condition may explain the blocking phenomenon which exists in the atmosphere and ocean and generated by topographic forcing.
01 Jan 2013
TL;DR: In this article, the Laplace transform of Euler-Cauchy equation with variable coefficients is used to solve the differential equation with free/forced oscillations and variable coefficients.
Abstract: It is well known fact that the Laplace transform is useful in solving linear ordinary differential equations with constant coefficients such as free/forced oscillations, but in the case of differential equation with variable coefficients is not. In here, we would like to propose the Laplace transform of Euler-Cauchy equation with variable coefficients, and findthe solution of Euler-Cauchy equation represented by the differential operator using Laplace transform. The purpose of this research is to make an application to its differ- ence equation and oscillation.
Journal Article•
TL;DR: In this paper, the sinecosine function was used to construct the traveling wave solutions for three models; namely, the (2+1)-dimensional Zoomeron equation, the Duffing equation and the Symmetric Regularized Long Wave equation (SRLW).
Abstract: In this paper, we use the sine-cosine function method to construct the traveling wave solutions for three models; namely the (2+1)-dimensional Zoomeron equation, the Duffing equation and the Symmetric Regularized Long Wave equation (SRLW). These equations play a very important role in mathematical physics and engineering sciences.
TL;DR: In this article, a new boundary-domain integral equation for convective heat transfer problems is derived, which accounts for fluid velocity which is computed using the velocity boundarydomain integral equations presented in [2], [3] (X.W. Gao, 2004, 2005).
TL;DR: In this paper, a Fokker-Planck equation on fractal curves is obtained, starting from Chapmann-Kolmogorov equation on the fractal curve.
Abstract: A Fokker–Planck equation on fractal curves is obtained, starting from Chapmann–Kolmogorov equation on fractal curves. This is done using the recently developed calculus on fractals, which allows one to write differential equations on fractal curves. As an important special case, the diffusion and drift coefficients are obtained, for a suitable transition probability to get the diffusion equation on fractal curves. This equation is of first order in time, and, in space variable it involves derivatives of order α, α being the dimension of the curve. An exact solution of this equation with localized initial condition shows departure from ordinary diffusive behavior due to underlying fractal space in which diffusion is taking place and manifests a subdiffusive behavior. We further point out that the dimension of the fractal path can be estimated from the distribution function.
04 Jun 2013
TL;DR: In this article, a rigorous equation is stated and it is shown that the spatial derivative of the Cole-Hopf solution of the KPZ equation is a solution of this equation, and the method of proof used to show that a process solves this equation is based on rather weak estimates.
Abstract: A rigorous equation is stated and it is shown that the spatial derivative of the Cole–Hopf solution of the KPZ equation is a solution of this equation The method of proof used to show that a process solves this equation is based on rather weak estimates so that this method has the advantage that it could be used to verify solutions of other highly singular SPDEs, too
TL;DR: In this paper, the iterative method developed by Daftardar-Gejji and Jafari (DJ method) is employed for analytic treatment of Laplace equation with Dirichlet and Neumann boundary conditions.
Abstract: In this paper, the iterative method developed by Daftardar-Gejji and Jafari (DJ method) is employed for analytic treatment of Laplace equation with Dirichlet and Neumann boundary conditions. The method is demonstrated by several physical models of Laplace equation. The obtained results show that the present approach is highly accurate and requires reduced amount of calculations compared with the existing iterative methods.
TL;DR: The iterative method for numerically solving the nonlinear Volterra-Fredholm integral equation is presented and the approximate solutions of integral equation are obtained.
TL;DR: A new method is presented which is based on the Taylor Matrix Method to give approximate solution of the linear fractional Fredholm integro-differential equations.
Abstract: In this article, we present a new method which is based on the Taylor Matrix Method to give approximate solution of the linear fractional Fredholm integro-differential equations. This method is based on first taking the truncated Taylor expansions of the functions in the linear fractional differential part and Fredholm integral part then, substituting their matrix forms into the equation. We solve this matrix equation with the assistance of Maple 13. In addition, illustrative examples are presented to demonstrate the effectiveness of the proposed method.