using triangular orthogonal functions for solving fredholm integral equations of the second kind the present work proposes a method for solving fredholm integral equations this is demonstrated by using a complementary pair of orthogonal triangular functions set derived from the well known block pulse functions set the operational matrices for integration product of two triangular functions and some formulas for calculating definite integral of them are derived and utilized to reduce the solution of fredholm integral equation to the solution of algebraic equations illustrative examples are included to show the high accuracy of the estimation and to demonstrate validity and applicability of the method

characterisation of bifurcation from the continuous spectrum by nodal properties where b is a densely defined linear operator in some banach space x with a purely continuous spectrum and f is a nonlinearity of higher order terms so that u 0 is a trivial solution for all l e r we recall that a number 1 is called a bifurcation point for 1 1 j if there are nontrivial solutions u 1 to 1 1 in each neighbourhood of 0 1 in x x r problems whose linearizations have a purely continuous spectrum typically arise in the study of nonlinear elliptic eigenvalue problems on zrrzbourzded domains see benci fortunato 3 berestycki lions 4 berger 5 bongers heinz kiipper 8 chiapinelli stuart 9 drager 111 ktipper 18 191 stuart 24 271 toland 28 or in the study of nonlinear integral equations of convolution type see amick toland 11 demay lo in these papers it has been established that the lowest point of the continuous spectrum is a bifurcation point if the nonlinearity f provides a sufficiently strong perturbation of the linear part b a number of different

higher holder regularity for the fractional p laplacian in the superquadratic case abstract we prove higher holder regularity for solutions of equations involving the fractional p laplacian of order s when p 2 and 0 s 1 in particular we provide an explicit holder exponent for solutions of the non homogeneous equation with data in l q and q n s p which is almost sharp whenever s p p 1 n q the result is new already for the homogeneous equation

a chebyshev approximation for solving nonlinear integral equations of hammerstein type abstract a numerical method for solving fredholm volterra hammerstein integral equations is presented this method is based on replacement of the unknown function by truncated series of well known chebyshev expansion of functions the quadrature formula which we use to calculate integral terms can be estimated by fast fourier transform fft the numerical examples and the number of operations show the advantages of this method to some other usual methods

new approach for numerical solution of hammerstein integral equations in this paper we use a numerical procedure for solving a class of nonlinear integral equations of the hammerstein type using the globally defined sinc basis functions properties of the sinc procedure are utilized to reduce the computation of the hammerstein equations to some algebraic equations we used some numerical examples to illustrate the accuracy and the implementation of the method

numerical implementation of adomian decomposition method for linear volterra integral equations of the second kind in this paper we propose new ideas to implement adomian decomposition method to solve volterra integral equations numerical examples are presented to illustrate the method for linear volterra integral equations of the second kind

an expansion method to treat integral equations this paper introduces a method for treating integral equation of the second and first kind of both types fredholm or volterra the method is very useful in treating integral equation of the first kind the method is tested by various numerical examples

nodal properties and variational characterizations of solutions to nonlinear sturm liouville problems on considere des problemes aux valeurs propres non lineaires pour des equations differentielles de la forme p t u q t u f t u r t u avec des conditions aux limites de dirichlet

application of sinc collocation method for solving a class of nonlinear fredholm integral equations in this paper the numerical solution of nonlinear fredholm integral equations of the second kind is considered by two methods the methods are developed by means of the sinc approximation with the single exponential se and double exponential de transformations these numerical methods combine a sinc collocation method with the newton iterative process that involves solving a nonlinear system of equations we provide an error analysis for the methods so far approximate solutions with polynomial convergence have been reported for this equation these methods improve conventional results and achieve exponential convergence some numerical examples are given to confirm the accuracy and ease of implementation of the methods

using rationalized haar wavelet for solving linear integral equations the main purpose of this paper is to demonstrate that using rationalized haar wavelet for solving linear fredholm integral equation of the second kind we convert the integral equation to a system of linear equations and by using numerical examples we show our estimation have a good degree of accuracy

extremal constant sign solutions and nodal solutions for the fractional p laplacian abstract we study a pseudo differential equation driven by the degenerate fractional p laplacian under dirichlet type conditions in a smooth domain first we show that the solution set within the order interval given by a sub supersolution pair is nonempty directed and compact hence endowed with extremal elements then we prove existence of a smallest positive a biggest negative and a nodal solution combining variational methods with truncation techniques

the conditioning of linear boundary value problems investigation is made into the sensitivity of solutions of linear boundary value problems to perturbations of the boundary condition we derive a useful quantity to decide for well or ill conditioning from this it is deduced which kind of requirements the boundary conditions should meet in order to have a well conditioned problem it is shown that this quantity can fruitfully be used to explain why e g the multiple shooting technique is stable in contrast to the single shooting one for certain well posed problems having solutions with different growth behavior the results are sustained by a number of examples

solution of a volterra integral equation by the sinc collocation method in this paper a collocation procedure is developed for a linear volterra integral equation of the second kind by using the sinc basis functions the approximate solutions are given so that the auxiliary basis functions satisfy the four different end conditions it is shown that the sinc procedure converges to the solution at an exponential rate of o e k n numerical results are included to confirm the efficiency and accuracy of the method

numerical solutions for the system of fredholm integral equations of second kind by a new approach involving semiorthogonal b spline wavelet collocation method in this paper a new approach based on linear b spline wavelet method has been developed to approximate the solutions of system of linear fredholm integral equations of second kind compactly supported linear semi orthogonal b spline scaling functions and wavelet functions together with their dual functions are applied to approximate the solutions of linear fredholm integral equations system of second kind this method reduces the system of integral equations to a linear system of algebraic equations that can be solved easily with any of the usual methods the numerical results obtained by the present method have been compared with those obtained by adaptive method based on trapezoidal rule numerical examples are presented to illustrate the accuracy of the method

existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains abstract we consider nonlinear elliptic eigenvalue problems on unbounded domains g r n using an extended ljusternik schnirelman theory we prove the existence of infinitely many eigenfunctions on every sphere in l 2 g moreover we establish that the infimum of the spectrum of the linearized problem l is always a bifurcation point in addition there is an infinity of branches emanating at from the trivial line of solutions if belongs to the essential spectrum of l

a hopf s lemma and a strong minimum principle for the fractional p laplacian abstract our propose here is to provide a hopf lemma and a strong minimum principle for weak supersolutions of p s u c x u p 2 u in where is an open set of r n s 0 1 p 1 c c and p s is the fractional p laplacian

the stability of one step schemes for first order two point boundary value problems the stability of a finite difference scheme is related explicitly to the stability of the continuous problem being solved at times this gives materially better estimates for the stability constant than those obtained by the standard process of appealing to the stability of the numerical scheme for the associated initial value problem

numerical solution of some classes of integral equations using bernstein polynomials this paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using bernstein polynomials as basis the integral equations considered are fredholm integral equations of second kind a simple hypersingular integral equation and a hypersingular integral equation of second kind the method is explained with illustrative examples also the convergence of the method is established rigorously for each class of integral equations considered here

properties of eigenvalues and some regularities on fractional p laplacian with singular weights abstract we provide fundamental properties of the first eigenpair for fractional p laplacian eigenvalue problems under singular weights which is related to hardy type inequality and also show that the second eigenvalue is well defined we obtain a priori bounds and the continuity of solutions to problems with such singular weights with some additional assumptions moreover applying the above results we show a global bifurcation emanating from the first eigenvalue the fredholm alternative for non resonant problems and obtain the existence of infinitely many solutions for some nonlinear problems involving singular weights these are new results even for fractional laplacian

on optimal shooting intervals we develop an adaptive multiple shooting strategy which is nearly optimal with respect to cpu time since the costs of integration are the most important components in this we investigate in some detail how the gridpoints are chosen by an adaptive integration routine we use this information to find out where the shooting points have to be selected we also show that our final strategy is stable in the sense that rounding errors can be kept below a given tolerance finally we pay attention to the question how the need for memory can be minimized

two methods for solving integral equations the objective of this work is to introduce two efficient strategies that produce the accurate solutions to linear and nonlinear volterra and fredholm integral equations of the second kind for illustration several linear nonlinear and integro differential examples have been tested and the results showed improvements over existing techniques

collocation method for linear and nonlinear fredholm and volterra integral equations a collocation procedure is developed for the linear and nonlinear fredholm and volterra integral equations using the globally defined b spline and auxiliary basis functions the solution is collocated by cubic b spline and then the integral equation is approximated by the 5 points gauss turan quadrature formula with respect to the legendre weight function combination of these two approaches is the main idea of this paper to reduce the cost of computation and complexity the error analysis of proposed numerical method is studied theoretically numerical results are given to illustrate the efficiency of the proposed method the results are compared with the results obtained by other methods to verify that this method is accurate and efficient

p laplace equations with singular weights we study a class of p laplacian dirichlet problems with weights that are possibly singular on the boundary of the domain and obtain nontrivial solutions using morse theory in the absence of a direct sum decomposition we use a cohomological local splitting to get an estimate of the critical groups

existence and some properties of solutions for degenerate elliptic equations with exponent variable abstract in this paper we study degenerate elliptic equations with variable exponents when a perturbation term satisfies the ambrosetti rabinowitz condition and does not satisfy the ambrosetti rabinowitz condition for the first case we employ the standard mountain pass theorem to give the existence of solutions for the second case we use browder s theorem for monotone operators to show the unique existence of a solution when the perturbation term is decreasing with respect to a function variable a priori bound and nonnegativeness of solutions are also given we emphasize that the log holder continuous condition is not required

accurate estimates for the fundamental solutions of discrete boundary value problems it is investigated how the solutions of a discretized ode can be related to solutions of the ode by using a perturbation analysis based on boundary value formulations in particular it is indicated how one can find estimates for discrete modes that correspond to certain continuous modes these estimates are fairly sharp in a relative sense as is shown by some examples as a consequence one can deduce that certain properties of the one solution space like a dichotomy carry over to the other

cubic b spline approximation for linear stochastic integro differential equation of fractional order abstract in this paper the cubic b spline collocation method is used for solving the stochastic integro differential equation of fractional order we show that stochastic integro differential equation of fractional order is equivalent to a modified stochastic integral equation then we apply the proposed method to obtain a numerical scheme of the modified stochastic integral equation using this method the problem solving turns into a linear system solution of equations also the convergence analysis of this numerical approach has been discussed in the end examples are given to test the accuracy and the implementation of the method the results are compared with the results obtained by other methods to verify that this method is accurate and efficient

unique solvability of nonlinear volterra equations in weighted spaces abstract we investigate integral equations of the form x t g t t f t s x s ds in general this equation is history dependent so one needs to give an initial condition on 0 in order to obtain a unique solution by introducing a weight function on r we can single out a class of admissible solutions and give conditions for the unique solvability of in this restricted class we also study some fredholm equations on these weighted spaces in addition we also treat a class of equations of the first kind for which similar conclusions can be drawn

numerical solution of linear integral equations by using sinc collocation method a collocation procedure is developed for linear fredholm integral equations of the second kind using sinc basis functions the convergence rate of the method is o e k m numerical results are included to confirm the efficiency and accuracy of the method

solving nonlinear integral equations in the urysohn form by newton kantorovich quadrature method the newton kantorovich method is a well known method for solving nonlinear integral equations this method attempts to solve a sequence of linear integral equations in this article we develop a new method which is a combination of the newton kantorovich and quadrature methods the new method solves the nonlinear integral equations of the urysohn form in a systematic procedure some numerical examples are provided and the obtained numerical approximations are compared with the corresponding exact solutions

harmonic wavelet method towards solution of the fredholm type integral equations of the second kind periodic harmonic wavelets phw were applied as basis functions in solution of the fredholm integral equations of the second kind two equations were solved in order to find out advantages and disadvantages of such choice of the basis functions it is proved that phw satisfy the properties of the multiresolution analysis