an extrapolation method with step size control for nonlinear volterra integral equations in 6 it has been shown that the midpoint rule applied to second kind volterra integral equations possesses an asymptotic expansion in even powers of the stepsizeh in this paper we describe an extrapolation method based on the midpoint rule together with a mechanism of step size control

parabolic monge ampere equations on riemannian manifolds on a compact riemannian manifold m g we consider the parabolic monge ampere equation t x t logdet g x hess x t detg x x t f x x 0 phiv 0 x here is a real parameter andf 0 m r are smooth functions we show existence of for all timestindependent of if 0 then t t converges exponentially towards a solution of the stationary problem ast in the special case 0 f 0 one has 0 and we determine the convergence rate t in thel2 norm more precisely

existence of the global solution for the parabolic monge ampere equations on compact riemannian manifolds abstract on a compact riemannian manifold m g we consider the existence and nonexistence of global solutions for the parabolic monge ampere equation t log det g hess det g p f x x 0 0 x here p 1 and are real parameters f 0 m 0 are smooth functions on m if 0 then the solution of exists for all times t and t t converges exponentially towards a solution of its stationary equation as t in the case of 0 it does not have the global solution of thus we obtain the nonexistence of the positive solution for the stationary equation of

resolvents and solutions of singular volterra integral equations with separable kernels the volterra integral equation 1 x t a t 0 tb t s x s dswith a kernel of the form b t s p t q s is investigated where a p and q are functions that are defined a e on an interval 0 t and are measurable the main result of this paper states that if qa is lebesgue integrable on 0 t the sign of b t s does not change for almost all t s and if there is a function f that is continuous on 0 t except possibly at countably many points with b t t f t a e on 0 t then the function x defined by 2 x t a t 0 tr t s a s ds where 3 r t s b t s e s t b u u d u solves 1 a e on 0 t three diverse examples illustrate the efficacy of using 2 and 3 to calculate solutions of 1 two of the examples involve singular kernels the solution of one of them is nowhere continuous on 0 t

numerical methods for a volterra integral equation with non smooth solutions we consider the numerical treatment of a singular volterra integral equation with an infinite set of solutions one of which is smooth and all others have infinite gradient at the origin this equation has been the subject of previous works where we have dealt with the approximation of the smooth solution here we present numerical methods which enable us to obtain approximations to any of the infinite class of solutions some numerical examples are given which illustrate the performance of the methods employed

rigorous convergence analysis of jacobi spectral galerkin methods for volterra integral equations with noncompact kernels abstract this article deals with the implementation of jacobi spectral and pseudospectral galerkin approaches for solving weakly singular volterra integral equations with noncompact kernels the computational domain of the main problem and intervals of the involved integrals are transformed into 1 1 by some appropriate change of variables therefore one can apply the jacobi galerkin method and impose the inner product conditions on the residual of the considered equation also jacobi gauss quadrature rules are implemented to approximate the integral operator and the inner products associated to the jacobi weight function in the pseudospectral galerkin method for both cases of spectral and pseudospectral galerkin schemes rigorous convergence analysis is provided under some mild conditions on the smoothness of kernel and source functions in l and weighted l 2 norms several numerical examples are also given to confirm the theoretical prediction

global existence of solutions to parabolic monge ampere equations on riemannian manifolds abstract in this paper we study the cauchy problem of the parabolic monge ampere equation u t log g 1 det g i j i j u n log u in m n 0 u x 0 u 0 x in m n where m n is a compact complete riemannian manifold of dimension n 2 g i j denotes the metric of m n g det g i j 0 and u 0 x 1 is a smooth function we prove the global existence of solutions

variable transformations in the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels extensions to fredholm integral equations abstract a number of techniques that use variable transformations in numerical integration have been developed recently cf sidi numerical integration iv h brass g hammerlin eds birkhauser basel 1993 pp 359 373 laurie j comput appl math 66 1996 337 344 the use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule in this paper the application of variable transformation techniques of sidi and laurie type to the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels is considered since the transformations are such that the end points of integration need not be used as mesh points the methods introduced can be used for vie with both continuous and weakly singular kernel in a uniform way the methods have also the advantages of simplicity of application and of achieving high order of convergence the application of the idea to fredholm integral equations with continuous and weakly singular equations is also considered numerical results are included and they verify the expected increased order of convergence they were obtained by using the trapezoidal formula for the evaluation of the transformed integrals

spectral collocation methods for volterra integro differential equations with noncompact kernels the aim of this paper is to analyze spectral collocation methods for a class of volterra integro differential equations with noncompact kernels the spectral collocation method is proposed to solve the equations after the results on existence uniqueness and regularity of the solutions are established the convergence rates of the methods are proven and a numerical example is provided to verify the theoretical findings

existence and non existence of solutions for a class of monge ampere equations abstract we study the boundary value problems for monge ampere equations det d 2 u e u in r n n 1 u 0 first we prove that any solution on the ball is radially symmetric by the argument of moving plane then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution and vice versa using the comparison between domains we can prove that this phenomenon occurs for every domain finally we consider an equivalent problem with a parameter det d 2 u e t u in u 0 t 0 by using lyapunov schmidt reduction method we get the local structure of the solutions near a degenerate point by leray schauder degree theory a priori estimate and bifurcation theory we get the global structure

the exact solution of a class of volterra integral equation with weakly singular kernel abstract in this paper the weakly singular volterra integral equations with an infinite set of solutions are investigated among the set of solutions only one particular solution is smooth and all others are singular at the origin the numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation the aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory applying weighted integral we provide a new definition for reproducing kernel space and obtain reproducing kernel function using the good properties of reproducing kernel function the only smooth solution is exactly expressed in the form of series the n term approximate solution is obtained by truncating the series meanwhile we prove that the derivative of approximation converges to the derivative of exact solution uniformly the final numerical examples compared with other methods show that the method is efficient

extrapolation method of iterated collocation solution for two dimensional nonlinear volterra integral equations in this paper we study the numerical solution of two dimensional nonlinear volterra integral equation by collocation and iterated collocation method asymptotic error expansion of iterated collocation solution is obtained using richardson s extrapolation we show that the accuracy of numerical solution in the iterated collocation method can be improved greatly some numerical examples are given to illustrate this theory

numerical analysis of a singular integral equation in this paper we give details of the numerical analysis of an integral equation of the form 1 y t f t email protected 0ts m 1t my s email protected 0 the distinctive feature of the equation is the presence of a singularity at t 0 for all values of m 0 and at s 0 for all values of t 0 for 0 m 1 this means that conventional analytical and numerical theory does not apply in fact for 0 m 1 the equation has an infinite family of solutions as background we give details of the analytical results on existence and uniqueness of solutions to 1 and we give new results on the use of numerical schemes that will yield approximations to any specific solution we conclude with some numerical results that show that our methods enable us to find approximations of any chosen order to any of the infinite class of exact solutions to 1

volterra integral equations and a new gronwall inequality part ii the nonlinear case we consider nonlinear singular volterra integral equations of the second kind we generalise the transformation method introduced in part i of this paper 6 to cope with both the nonlinearity and slightly more general singular kernels we also consider a particular class of nonlinear equation for which the solution behaviour is known using this a priori knowledge we propose a modification of the transformation technique which results in a numerical method with good asymptotic stability properties applying the general theory of part i of this paper we prove convergence of this scheme

some existence results for fully nonlinear elliptic equations of monge ampere type we study the existence and uniqueness of solutions of monge ampere type equations this type of equations has been studied extensively by caffarelli nirenberg spruck and many others see 5 through 8 and the references therein we present some existence and uniqueness results for this type of equations on compact riemannian manifolds with non negative sectional curvature we have also generalized some results in 7

existence and uniqueness of solutions of abel integral equations with power law non linearities abstract in this article a class of nonlinear abel type integral equations is studied these equations have at least the trivial solution but we are interested in a continuous positive solution we show that there exists a unique positive solution in an order interval of a cone in a banach space

convergence analysis of spectral collocation methods for a class of weakly singular volterra integral equations abstract in this paper we discuss the application of spectral jacobi collocation methods to a certain class of weakly singular volterra integral equations first we use some function transformations and variable changes to transform the equation into a volterra integral equation defined on the standard interval 1 1 then the jacobi gauss quadrature formula is used to approximate the integral operator for the spectral jacobi collocation method a rigorous error analysis in both the l and weighted l 2 norms is given under the assumption that both the kernel function and the source function are sufficiently smooth finally some numerical examples are provided to illustrate the theoretical results

equations du type monge ampere sur les varietes riemanniennes compactes i let vn g be a c compact riemannian manifold for a suitable function on vn let us consider the change of metric g g hess and the function as a ratio of two determinants m g g 1 using the method of continuity we first solve in c the problem log m 0 c then under weak hypothesis on f we solve the general equation log m f p f in c vn using a method of iteration our study gives rise to an interesting a priori estimate on which does not occur in the complex case this estimate should enable us to solve the equation above when 0 providing we can overcome difficulties related to the invertibility of the linearised operator this open question will be treated in our next article

numerical solutions of weakly singular volterra integral equations using the optimal homotopy asymptotic method the purpose of this paper is to apply a numerical technique namely the optimal homotopy asymptotic method oham for finding the approximate solutions of a class of volterra integral equations with weakly singular kernels this method uses simple computations with quite acceptable approximate solutions which has close agreement with exact solutions illustrative examples are included to demonstrate the validity and applicability of the present method and a comparison has been made with existing results

equations du type de monge ampere sur les varietes riemanniennes compactes ii abstract let vn g be a c compact riemannian manifold without boundary given the following changes of metric g g hess l 2 g g 2 hess where a is a fixed constant we study the corresponding monge ampere equations 1 log g g 1 f p 2 log g g 1 f p we first solve eq 2 under some simple assumptions on f c then using an appropriate change of functions that enables us to take advantage of the estimates just carried out for eq 2 we extend to eq 1 all the results proved in our previous articles 5 6 for the usual monge ampere equation although equation 2 is not locally invertible and does not even admit a solution for all f 0 f c v n a similar change of functions leads to partial results about eq 1 via c2 and c3 estimates for eq 2 eventually we give some comments and errata of our previous article p delanoe j funct anal 41 1981 341 353

numerical solution of a nonuniquely solvable volterra integral equation using extrapolation methods in this work the numerical solution of a volterra integral equation with a certain weakly singular kernel depending on a real parameter is considered although for certain values of this equation possesses an infinite set of solutions we have been able to prove that euler s method converges to a particular solution it is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize so that general extrapolation algorithms like the e algorithm can be applied to improve the numerical results this is illustrated by means of some examples

collocation methods for second kind volterra integral equations with weakly singular kernels in this paper it is shown that the use of uniform meshes leads to optimal convergence rates provided that the analytic solutions of a particular class of volterra integral equations vies are smooth if the exact solutions are not smooth however suitable transformations can be made so that the new vies possess smooth solutions spline collocation methods with uniform meshes applied to these new vies are then shown to be able to yield optimal global convergence rates the general theory is applied to a typical case i e the integral kernels consisting of the singular term t s

poisson transforms on some c algebras generated by isometries abstract a noncommutative poisson transform associated to a certain class of sequences of operators on hilbert spaces with property p is defined on some universal c algebras resp nonselfadjoint algebras generated by isometries its properties are described and used to study these universal algebras and their representations as consequences we obtain a functional calculus isometric resp unitary dilations and commutant lifting theorem for the class of sequences of operators with property p our geometrical approach leads also to new and elementary proofs as well as extensions of some classical results

existence of solutions to initial value problem for a parabolic monge ampere equation and application abstract for the initial value problem of the parabolic monge ampere equation v t v xx rxv x v xx v x 2 0 x t r 0 t arising from the optimal investment of mathematical finance theory we establish the existence of solutions whose application is also given here the initial function is unbounded and a special property is required for the solution to satisfy

product integration methods for second kind abel integral equations the construction and convergence of high order product integration methods for the second kind abel equation are discussed and the results of de hoog and weiss are generalised backward difference methods are introduced and numerical results are presented which verify the theoretical rates of convergence