a difference scheme for a nonlinear partial integrodifferential equation a difference method for the numerical integration of a nonlinear partial integrodifferential equation is considered the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in proving consistency the verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature a stability result is derived that is applicable to equations and numerical methods far more general than those treated in this paper

a semi discrete scheme for solving nonlinear hyperbolic type partial integro differential equations using radial basis functions in this paper we propose an effective numerical method for solving nonlinear volterra partial integro differential equations these equations include the partial differentiations of an unknown function and the integral term containing the unknown function as the memory of system radial basis functions and finite difference method as the main techniques play the important role to reduce a nonlinear partial integro differential equation to a linear system of equations some examples are demonstrated to describe the method numerical results confirm the validity and efficiency of the presented method

spline collocation methods for a class of hyperbolic partial integro differential equations spline collocation methods are proposed for the spatial discretization of a class of hyperbolic partial integro differential equations arising in the theory of linear viscoelasticity for problems in two space variables error estimates are derived for the continuous time orthogonal spline collocation method and three discrete time orthogonal spline collocation methods the continuous time modified cubic spline collocation method for problems in one space variable is also analyzed

numerical analysis of volterra integro differential equations for viscoelastic rods and membranes abstract we consider the initial boundary value problems for a homogeneous volterra integro differential equations for viscoelastic rods and membranes in a bounded smooth domain the memory kernel of the equation is made up in a complicated way from the distinct moduli of stress relaxation for compression and shear which is challenging to approximate the literature reported on the numerical solution of this model is extremely sparse in this paper we will study the second order continuous time galerkin approximation for its space discretization and propose a fully discrete scheme employing the crank nicolson method for the time discretization then we derive the uniform long time error estimates in the norm l t 1 0 l x 2 for the finite element solutions some numerical results are presented to illustrate our theoretical error bounds

minimal and maximal methods of interpolation abstract we show that several interpolation functors including the widely used real and complex methods are minimal or maximal extensions from a single couple of banach spaces various consequences are drawn from this property

a second order accurate numerical method for a fractional wave equation we study a generalized crank nicolson scheme for the time discretization of a fractional wave equation in combination with a space discretization by linear finite elements the scheme uses a non uniform grid in time to compensate for the singular behaviour of the exact solution at t 0 with appropriate assumptions on the data and assuming that the spatial domain is convex or smooth we show that the error is of order k2 h2 where k and h are the parameters for the time and space meshes respectively

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

uniform l1 error bounds for the semidiscrete solution of a volterra equation with completely monotonic convolution kernel abstract we study the second order continuous time galerkin approximation of the solution ut 0t t s au s ds 0 u 0 v t 0 where a is an elliptic partial differential operator and t is completely monotonic and locally integrable but not constant error estimates are derived in the norm of lt1 0 lx2

a numerical method for a partial integro differential equation a method is considered for the integration in time of a partial integro differential equation the discretization technique employed is patterned after an idea of ch lubich error bounds are derived for both smooth and nonsmooth initial data

quasi wavelet based numerical method for volterra integro differential equations on unbounded spatial domains in this paper we study the numerical solution of the parabolic volterra integro differential equation with a memory term on one dimensional unbounded domain a new numerical method is proposed by applying the forward euler scheme in time and the quasi wavelet based numerical scheme in space some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method

alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive type memory term new numerical techniques are presented for the solution of a class of linear partial integro differential equations pides with a positive type memory term in the unit square in these methods orthogonal spline collocation osc is used for the spatial discretization and for the time stepping new alternating direction implicit adi methods based on the backward euler the crank nicolson and the second order bdf methods combined with judiciously chosen quadrature rules are considered the adi osc methods are proved to be of optimal accuracy in time and in the l 2 norm in space numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the l infty and h 1 norms and superconvergence phenomena

convergence and rate of approximation for linear integral operators in bvφ spaces in multidimensional setting abstract in this paper where for the first time it is introduced a multidimensional concept of variation in the sense of tonelli we extend previous results concerning convergence order of approximation and higher order of approximation for linear integral operators in bv r n space of functions with bounded variation in r n moreover we give a further generalization of the theory introducing the concept of f variation where f is a continuous sublinear functional

a parallel method for the numerical solution of integro differential equation with positive memory abstract an efficient parallel numerical method is proposed for an integro differential equation with positive memory instead of solving the equation in classical time marching methods which require massive storage of solutions of previous time steps in order to advance to a next time step the fourier laplace transformation in time is applied to obtain a set of complex valued elliptic problems parameterized by points on a contour in the complex plane using the independence of an elliptic problem corresponding to one contour point is independent of those elliptic problems corresponding to other contour points all elliptic problems can be solved in parallel essentially without data communications then the time domain solution can be obtained by the fourier laplace inversion formula an error analysis and the numerical implementation of this parallel method is presented

orthogonal spline collocation method for the fourth order diffusion system abstract the fourth order diffusion systems depict the wave and photon propagation in intense laser beams and play an important role in the phase separation in binary mixture in this paper by using orthogonal spline collocation osc method in spatial direction and classical l1 approximation in temporal direction a fully discrete scheme is established for a class of fourth order fractional reaction diffusion equations for the original unknown u and auxiliary variable v u the full discrete unconditional stabilities based on a priori analysis are derived by virtue of properties of osc moreover the convergence rates in l 2 norm for unknown u are strictly investigated at the same time the optimal error estimates in h 1 norm for unknown u and in l 2 norm for variable v are also derived respectively for further verifying the theoretical analysis some numerical examples are provided

the integration theory of banach space valued measures and the tonelli fubini theorems ii pettis integration in this part of the paper we turn to the concepts of gelfand and pettis integrability and integration of scalar valued functions with respect to a banach space valued measure the problem before us and our method for solving it have been outlined in section 1 of part i of this paper in what follows we adhere to the notation of part i and continue the trend set in that part all references with numbers such as 1 2 2 3 and a5 are to the items in sections 1 and 2 and appendix a in part i

vitali type theorems for filter convergence related to vector lattice valued modulars and applications to stochastic processes a vitali type theorem for vector lattice valued modulars with respect to filter convergence is proved some applications are given to modular convergence theorems for moment operators in the vector lattice setting and also for the brownian motion and related stochastic processes

complex interpolation of lp spaces of vector measures on δ rings abstract we apply the calderon interpolation methods to banach lattices of p integrable and weakly p integrable functions with respect to a banach space valued measure defined on a ring in general the results we obtain are quite different from those in the case of vector measures on algebras however we find a wide class of vector measures on rings for which the results on algebras hold true

the max product generalized sampling operators convergence and quantitative estimates abstract in this paper we study the max product version of the generalized sampling operators based upon a general kernel function in particular we prove pointwise and uniform convergence for the above operators together with a certain quantitative jackson type estimate based on the first order modulus of continuity of the function being approximated the proof of the proposed results are based on the definition of the so called generalized absolute moments by the proposed approach the achieved approximation results can be applied for several type of kernels not necessarily duration limited such as the sinc function the fejer kernel and many others examples of kernels with compact support for which the above theory holds can be given for example by the well known central b splines

an h 1 galerkin mixed finite element method for an evolution equation with a positive type memory term an h1 galerkin mixed finite element method is analyzed for a class of evolution equations with memory when a classical mixed method is applied to such problems it has not been possible to obtain any estimate for the flux however the proposed approach yields optimal order convergence without the lbb consistency condition and quasi uniformity of the finite element mesh compared to the results proved for one space variable the l2 estimate of the flux is not optimal for problems in two and three space dimensions therefore a modification of the method is proposed and analyzed a maximum norm estimate is also derived in one and two space variables a backward euler approximation of the modified method is also analyzed

orthogonal spline collocation methods for some partial integrodifferential equations the laplace transform is introduced to analyze the method of orthogonal spline collocation for the semidiscretization of a class of linear partial integrodifferential equations arising in viscoelasticity problems for example the stability and convergence of the space discretization are examined