approximation of ill posed volterra problems via predictor corrector regularization methods first kind volterra problems arise in numerous applications from inverse problems in mathematical biology to inverse heat conduction problems unfortunately such problems are also ill posed due to lack of continuous dependence of solutions on data consequently numerical methods to solve first kind volterra equations are effective only when regularizing features are built into the algorithms or used to control stepsize classical methods often combine numerical discretization with tikhonov regularization but in doing so the underlying volterra or causal nature of the original problem is often destroyed instead a predictor corrector type of numerical method is proposed which combines at each step local regularization ideas with the use of small intervals of future data the result is a regularized numerical method which retains much of the causal nature of the volterra problem and may be solved in fast sequential steps often improving upon the performance of classical algorithms such as those

the second kind chebyshev wavelet method for solving fractional differential equations abstract in this paper the second kind chebyshev wavelet method is presented for solving linear and nonlinear fractional differential equations we first construct the second kind chebyshev wavelet and then derive the operational matrix of fractional order integration the operational matrix of fractional order integration is utilized to reduce the fractional differential equations to system of algebraic equations in addition illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method

nonpolynomial spline collocation for volterra equations with weakly singular kernels volterra integral equations of the second kind with weakly singular kernels possess in general solutions which are not smooth near the left endpoint of the interval of integration since ordinary polynomial spline collocation cannot lead to high order convergence we introduce special nonpolynomial spline spaces which are modelled after the structure of these solutions near the point of nonsmooth behavior collocation in these spaces will once more lead to high order methods analogous results are derived for volterra integro differential equations with weakly singular kernels

cas picard method for fractional nonlinear differential equation in this paper a computational method for solving the fractional nonlinear differential equation is introduced we proposed a method by utilizing the cas wavelets in conjunction with picard technique we call the method as cas picard method the fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by picard technique and then transformed into a system of algebraic equations by using the operational matrices of cas wavelets the error and supporting analysis of the method are also investigated the comparison analysis of method with other existing numerical methods is also performed

the general vector solutions of such linear coupled matrix fractional differential equations by using kronecker structures in this paper we demonstrate how kronecker products and vector operator can be combined beautifully to solve and fit the non homogenous linear matrix fractional differential equations and such coupled linear matrix fractional differential equations the way exists which transform the given coupled matrix fractional differential equations by using kronecker structures into forms for which solutions may be readily computed some important and interesting special cases of the general system of non homogeneous linear matrix fractional differential equations of fractional order are also considered which includes the non homogenous linear fractional dynamical system with delays in control finally a brief comparison between the fractional and integer order vector solutions and some examples are given to illustrate our new approaches

analytical solution of a special non homogeneous pyroelectric hollow cylinder for piezothermoelastic axisymmetric plane strain dynamic problems by virtue of the introduction of a dependant variable and the separation of variables technique the piezothermoelastic axisymmetric plane strain dynamic problem of a special non homogeneous pyroelectric hollow cylinder is transformed to a volterra integral equation of the second kind about a function with respect to time which can be solved successfully by means of the interpolation method then the solutions of displacements stresses electric displacements and electric potential are obtained the present method is suitable for a non homogeneous pyroelectric hollow cylinder with an arbitrary thickness subjected to arbitrary axisymmetric thermal loads numerical results are presented graphically

numerical solution for the variable order linear cable equation with bernstein polynomials abstract in this paper bernstein polynomials method is proposed for the numerical solution of a class of variable order fractional linear cable equation in this paper we adopted bernstein polynomials basis defined on the interval 0 r to solve the equations defined on the section 0 x 0 t the main characteristic behind this approach in this paper is that we derive two kinds of operational matrixes of bernstein polynomials with the operational matrixes the initial equation is transformed into the products of several dependent matrixes which can also be viewed as the system of linear equations after dispersing the variable by solving the linear system of algebraic equations the numerical solutions are acquired only a small number of bernstein polynomials are needed to obtain a satisfactory result numerical examples are provided to show that the method is computationally efficient

a nystrom interpolant for some weakly singular nonlinear volterra integral equations we consider a second kind weakly singular nonlinear volterra hammerstein integral equation defined by a compact operator and derive a nystrom type interpolant of the solution based on gauss radau nodes we prove the convergence of the interpolant and derive convergence estimates for equations with nonlinearity of algebraic kind we improve the rate of convergence by using a smoothing transformation some numerical examples are given

local regularization for the nonlinear inverse autoconvolution problem we develop a local regularization theory for the nonlinear inverse autoconvolution problem unlike classical regularization techniques such as tikhonov regularization this theory provides regularization methods that preserve the causal nature of the autoconvolution problem allowing for fast sequential numerical solution cal o rn 2 r 2n flops where r ll n for the method discussed in this paper as applied to the nonlinear problem in comparison the cost for tikhonov regularization applied to a general linear problem is cal o n 3 flops we prove the convergence of the regularized solutions to the true solution as the noise level in the data shrinks to zero and supply convergence rates for the cases of both l 2 and continuous data we propose several regularization methods and provide a theoretical basis for their convergence of note is that this class of methods does not require an initial guess of the unknown solution our numerical results confirm the effectiveness of the methods with results comparing favorably to numerical examples found in the literature for the autoconvolution problem e g g fleischer r gorenflo and b hofmann zamm z angew math mech 79 1999 pp 149 159 for examples using tikhonov regularization with total variation constraints and j janno inverse problems 16 2000 pp 333 348 for examples using the method of lavrent ev this especially seems to be true when it comes to the recovery of sharp features in the unknown solution we also show the effectiveness of our method in cases not covered by the theory

a hybrid collocation method for a nonlinear volterra integral equation with weakly singular kernel this work is concerned with the numerical solution of a nonlinear weakly singular volterra integral equation owing to the singular behavior of the solution near the origin the global convergence order of product integration and collocation methods is not optimal in order to recover the optimal orders a hybrid collocation method is used which combines a non polynomial approximation on the first subinterval followed by piecewise polynomial collocation on a graded mesh some numerical examples are presented which illustrate the theoretical results and the performance of the method a comparison is made with the standard graded collocation method

a wavelet approach for solving multi term variable order time fractional diffusion wave equation abstract we firstly generalize a multi term time fractional diffusion wave equation to the multi term variable order time fractional diffusion wave equation m v tfd e by the concept of variable order fractional derivatives then we implement the chebyshev wavelets cws through the operational matrix method to approximate its solution in the unit square in fact we apply the operational matrix of variable order fractional derivative omv fd of the cws to derive the unknown solution we proceed with coupling the collocation and tau methods to reduce m v tfd e to a system of algebraic equations the important privilege of method is handling different kinds of conditions i e initial boundary conditions and dirichlet boundary conditions by implementing the same techniques the convergence and error estimation of the cws expansion in two dimensions are theoretically investigated we also examine the applicability and computational efficiency of the new scheme through the numerical experiments

the piecewise polynomial collocation method for nonlinear weakly singular volterra equations second kind volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration using an adaptation of the analysis originally developed for nonlinear weakly singular fredholm integral equations we present a complete discussion of the optimal global and local order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear volterra integral equations with algebraic or logarithmic singularities in their kernels

a new approach to the numerical solution of weakly singular volterra integral equations we consider linear weakly singular volterra integral equations of the second kind with kernels of the form k x v x v k x v 0 1 or k x v log x v k x v k x v being a smooth function the solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration by a transformation of the unknown function we obtain an equation which is still weakly singular but whose solution is as smooth as we like this resulting equation is then solved by standard product integration methods

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

differential algebraic approach in non linear dynamical compensator design for d c to d c power converters abstract a non linear dynamical compensator design based on fliess generalized controller canonical form for non linear dynamical systems is proposed for the asymptotic stabilization of output load voltage or current in pulse width modulation controlled d c to d c power supplies the technique rests on the recently introduced differential algebraic approach to the study of controlled dynamical systems

the transient responses of a special non homogeneous magneto electro elastic hollow cylinder for axisymmetric plane strain problem by virtue of the introduction of new dependent variable and the separation of variables technique the transient responses of a special non homogeneous magneto electro elastic hollow cylinder are transformed to two volterra integral equations of the second kind of about two functions with respect to time these integral equations can be solved successfully by means of the interpolation method then the complete solutions of displacements stresses electric potential electric displacements magnetic potential and magnetic inductions are obtained the present method is suitable for a magneto electro elastic hollow cylinder with an arbitrary thickness subjected to arbitrary axisymmetric mechanical and electromagnetic loads numerical results are finally presented

on a graded mesh method for a class of weakly singular volterra integral equations in this paper a class of weakly singular volterra integral equations with an infinite set of solutions is investigated among the set of solutions only one particular solution is smooth and all others are singular at the origin the numerical solution of this class of equations has been a difficult topic to analyze and has received much previous investigation the aim of this paper is to improve the convergence rates by a graded mesh method the convergence rates are proved and a variety of numerical examples are provided to support the theoretical findings

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

collocation and iterated collocation methods for a class of weakly singular volterra integral equations we discuss the convergence properties of spline collocation and iterated collocation methods for a weakly singular volterra integral equation associated with certain heat conduction problems this work completes the previous studies of numerical methods for this type of equations with noncompact kernel in particular a global convergence result is obtained and it is shown that discrete superconvergence can be achieved with the iterated collocation if the exact solution belongs to some appropriate spaces some numerical examples illustrate the theoretical results

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

using qr factorization and svd to solve input estimation problems in structural dynamics input estimation problems in structural mechanics are in general ill posed treatment of these problems requires that the problems are reformulated usually by numerical regularization techniques several methods of solving input estimation problems which take the form of structured block matrix problems are studied in this paper the investigated methods are based on forming the normal equations qr factorization and singular value decomposition a specific block qr factorization algorithm utilizing the structure of the associated upper block triangular toeplitz matrix is also introduced moreover a criterion for choosing the level of regularization based on the data used to construct l curves is suggested numerical examples are given to illustrate various numerical aspects of the different methods and the performance of the advocated criterion

on the rotation of a non homogeneous composite infinite cylinder of orthotropic material abstract the rotation problem of an orthotropic cylinder containing i an isotropic core and ii a rigid core is considered the elastic constants and density are taken as a power function of the radial coordinate analytical expressions for the components of the stress in different cases are obtained finally the results are illustrated graphically

properties of interpolatory product integration rules this paper is concerned with the approximate evaluation of int 1 1 k x f x d x where k is lebesgue integrable and f is at least riemann integrable and preferably smooth the integral is approximated by a rule of the form sum nolimits i 1 n w ni k f x ni where the points x ni are chosen in some prescribed way and the weights w ni k are such that the rule is exact if f is any polynomial of degree n for suitable choices of the points and suitable functions k the rule is shown to converge to the exact integral and the companion rule sum w ni k f x ni to int k x f x dx for all riemann integrable functions f the companion property ensures for example that the weights have the asymptotic positivity property if k is positive error bounds are given that guarantee rapid convergence of the rule if f is smooth finally by establishing a connection with mean convergence of lagrangian interpolation some negative theorems are proved in pa

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

transient thermal stresses in a rotation non homogeneous cylindrically orthotropic composite tubes the present investigation deals with transient thermal stresses in a rotation non homogeneous cylindrically orthotropic composite tubes a solution is given for the stresses and displacement in rotation non homogeneous orthotropic hollow circular cylinder due to an imposed constant temperature on one surface and heat convection into a medium of different constant temperature at the other surface temperature independent material properties are assumed and a displacement approach is used numerical results were carried out for the temperature displacement and stresses

high order methods for weakly singular integral equations with nonsmooth input functions to solve one dimensional linear weakly singular integral equations on bounded intervals with input functions which may be smooth or not we propose to introduce first a simple smoothing change of variable and then to apply classical numerical methods such as product integration and collocation based on global polynomial approximations the advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well conditioned

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

a method for the numerical solution of the integro differential equations in this note the differential transformation is applied to solve the linear first order ordinary fredholm integro differential equations we will give an applicable relation between the one and two dimensional differential transformation in order to solve integro differential equations also we extend this method for searching the numerical solutions of linear higher order ordinary fredholm integro differential equations numerical examples are used to illustrate the preciseness and effectiveness of the proposed method

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

solving systems of fractional differential equations using differential transform method this paper presents approximate analytical solutions for systems of fractional differential equations using the differential transform method the fractional derivatives are described in the caputo sense the application of differential transform method developed for differential equations of integer order is extended to derive approximate analytical solutions of systems of fractional differential equations the solutions of our model equations are calculated in the form of convergent series with easily computable components some examples are solved as illustrations using symbolic computation the numerical results show that the approach is easy to implement and accurate when applied to systems of fractional differential equations the method introduces a promising tool for solving many linear and nonlinear fractional differential equations

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

bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations a new operational matrix for integration of bernoulli polynomials is introduced by using this new operational matrix of integration and the so called collocation method linear volterra and nonlinear volterra fredholm hammerstein integral equations are reduced to systems of algebraic equations with unknown bernoulli coefficients some error estimations are provided and illustrative examples are also included to demonstrate the efficiency and applicability of the technique a new operational matrix for integration of bernoulli polynomials is introduced a collocation method is used to reduce equations to systems of algebraic equations some error bounds and convergence results are provided for the proposed method the efficiency and applicability of the technique are checked by some examples

wavelets method for solving fractional optimal control problems in this paper an efficient and accurate computational method based on the legendre wavelets lws is proposed for solving a class of fractional optimal control problems focps in the proposed method the focp under consideration is reduced to a system of nonlinear algebraic equations which can be simply solved to this end the fractional derivative of the state variable and the control variable are expanded by the lws with unknown coefficients then the operational matrix of the riemann liouville fractional integration with some properties of the lws are employed to achieve a nonlinear algebraic equation in place of the performance index and a linear system of algebraic equations in place of the dynamical system in terms of the unknown coefficients finally the method of constrained extrema which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined lagrange multipliers is applied as a result the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable control variable and lagrange multipliers furthermore the efficiency of the proposed method is shown for some concrete examples the results reveal that the proposed method is very accurate and efficient

the matrix laguerre transform the laguerre transform introduced by keilson and nunn 1979 and keilson nunn sumita 1981 provides an algorithmic basis for the computation of multiple convolutions in conjunction with other algebraic and summation operations the methods enable one to evaluate numerically a variety of results in applied probability and statistics that have been available only formally for certain more complicated models the formulation must be extended in this paper we establish the matrix laguerre transform appropriate for the study of semi markov processes and markov renewal processes as an extension of the scalar laguerre transform the new formalism enables one to calculate matrix convolutions and other algebraic operations in matrix form as an application a matrix renewal function is evaluated and its limit theorem is numerically exhibited

high accuracy combination algorithm and a posteriori error estimation for solving the first kind abel integral equations abstract this paper presents high accuracy combination algorithm for solving the first kind abel integral equations to avoid solving ill posed problems we transform the first kind abel integral equation to the second kind volterra integral equation with a continuous kernel expressed by a weakly singular integral by using integration rules the approximation of this kernel can be easily computed then two quadrature algorithms for solving abel integral equations are proposed which possess accuracy order o h 1 0 h 2 moreover a posteriori error estimate for the algorithms is derived both theory and numerical examples show that it is effective and saves storage capacity and computational work

meshless methods for one dimensional oscillatory fredholm integral equations abstract in this paper efficient and simple algorithms based on levin s quadrature theory and our earlier work involving local radial basis function rbf and chebyshev differentiation matrices are adopted for numerical solution of one dimensional highly oscillatory fredholm integral equations this work is focused on the comparative performance of local rbf meshless and pseudospectral procedures we have tested the proposed methods on phase functions with and without stationary phase point s both on uniform and chebyshev grid points the proposed procedures are shown accurate and efficient and therefore provide a reliable platform for the numerical solution of integral equations from the numerical results we draw some conclusions about accuracy efficiency and robustness of the proposed approaches

legendre wavelets method for solving fractional partial differential equations with dirichlet boundary conditions in this paper a new method based on the legendre wavelets expansion together with operational matrices of fractional integration and derivative of these basis functions is proposed to solve fractional partial differential equations with dirichlet boundary conditions the proposed method is very convenient for solving such boundary value problems since the boundary conditions are taken into account automatically convergence of the two dimensional legendre wavelets expansion is investigated illustrative examples are included to demonstrate the validity and applicability of the presented wavelets method

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

a viscosity method with no spectral radius requirements for the split common fixed point problem abstract this paper is concerned with an algorithmic solution to the split common fixed point problem in hilbert spaces our method can be regarded as a variant of the viscosity approximation method under very classical assumptions we establish a strong convergence theorem with regard to involved operators belonging to the wide class of quasi nonexpansive operators in contrast with other related processes our algorithm does not require any estimate of some spectral radius the technique of analysis developed in this work is new and can be applied to many other fixed point iterations numerical experiments are also performed with regard to an inverse heat problem

closed form solutions to a class of h optimization problems a closed form solution is presented to the h sub optimal control problem via state or output feedback the problem formulation is general and does not impose any restrictions on feedthrough terms between the control input and the controlled output or between the disturbance input and the measurement output variables it is however required that both the transfer function from the disturbance input to the measurement output and the transfer function from the control input to the controlled output have no invariant zeros on the jw axis and that they satisfy certain geometric conditions for the same class of systems the conditions under which the h optimal control problem via state feedback has a solution are given moreover explicit expressions for the optimal solutions are provided finally the pole zero cancellations in the closed loop system resulting from the h optimal or sub optimal state or output feedback are examined

on haar wavelet operational matrix of general order and its application for the numerical solution of fractional bagley torvik equation abstract in the present analysis the motion of an immersed plate in a newtonian fluid described by torvik and bagley s fractional differential equation 1 has been considered this bagley torvik equation has been solved by operational matrix of haar wavelet method the obtained result is compared with analytical solution suggested by podlubny 2 haar wavelet method is used because its computation is simple as it converts the problem into algebraic matrix equation

an efficient numerical method for solving abel integral equation in this paper a numerical method for solving abel integral equation is presented we first convert the integral equation to the algebraic equation then we find numerical inversion of laplace transform by power series at last the pade approximants is effectively used to improve the convergence rate and accuracy of the computed series the method is described and illustrated with numerical examples the results reveal that the method is accurate and easy to implement

adomian decomposition method for solving fractional nonlinear differential equations in this article we have discussed a new application of adomian decomposition method on time fractional nonlinear fractional differential equations three models with fractional time derivative of order a 0 a 1 are considered and solved by means of adomian decomposition method the behaviour of adomian solutions and the effects of different values of a are investigated numerical examples are tested to illustrate the pertinent feature of the proposed algorithm

haar wavelet operational methods for the numerical solutions of fractional order nonlinear oscillatory van der pol system in this paper an efficient numerical method for the solution of nonlinear damped van der pol equation based on the haar wavelets approach is proposed the proposed scheme can be used for solving one of the widely studied and challenging equation in nonlinear dynamics like the damped van der pol equation we have applied a numerical procedure involving haar series for the highest order of derivative moreover the numerical result shows that the proposed mechanism is quite reasonable when compare to exact solution

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

scw method for solving the fractional integro differential equations with a weakly singular kernel in this paper based on the second chebyshev wavelets scw operational matrix of fractional order integration a numerical method for solving a class of fractional integro differential equations with a weakly singular kernel is proposed by using the operational matrix the fractional integro differential equations with weakly singular kernel are transformed into a system of algebraic equations the upper bound of the error of the second chebyshev wavelets expansion is investigated finally some numerical examples are shown to illustrate the efficiency and accuracy of the approach

meshless methods for two dimensional oscillatory fredholm integral equations abstract in this paper a meshless solution procedure incorporating delaminating quadrature method for two dimensional highly oscillatory fredholm integral equation is put forward the proposed method is an extension of our earlier findings of meshless methods for two dimensional oscillatory fredholm integral equation having kernel function free of stationary phase point s siraj ul islam et al 2015 the current method deals not only with the kernels having no stationary phase point s but also with the kernels having stationary phase point s in the context of highly oscillatory integral equations the new method is numerically stable and computationally fast numerical experiments are presented to testify the robustness and efficiency of the proposed method

numerical approach to differential equations of fractional order in this paper the variational iteration method and the adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order the two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations in these schemes the solution takes the form of a convergent series with easily computable components this paper presents a numerical comparison between the two methods for solving systems of fractional differential equations numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order

application of legendre wavelets for solving fractional differential equations in this paper we develop a framework to obtain approximate numerical solutions to ordinary differential equations odes involving fractional order derivatives using legendre wavelet approximations the properties of legendre wavelets are first presented these properties are then utilized to reduce the fractional ordinary differential equations fodes to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution

haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations haar wavelet operational matrix has been widely applied in system analysis system identification optimal control and numerical solution of integral and differential equations in the present paper we derive the haar wavelet operational matrix of the fractional order integration and use it to solve the fractional order differential equations including the bagley torvik ricatti and composite fractional oscillation equations the results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply

analysis of general quadrature methods for integral equations of the second kind this paper is concerned with a class of approximation methods for integral equations of the form y t f t int limits a b k t s y s ds wherea andb are finite f andy are continuous and the kernelk may be weakly singular the methods are characterized by approximate equations of the form y n t f t sum limits i 1 n w ni t y n s ni such methods include the nystrom method and a variety of product integration methods a general convergence theory is developed for methods of this type in suitable cases it has the feature that its application to a specific method depends only on a knowledge of convergence properties of the underlying quadrature rule the theory is used to deduce convergence results some of them new for a number of specific methods

on nonlinear control of euler lagrange systems disturbance attenuation properties abstract in this brief note we analyse the local disturbance attenuation properties of some asymptotically stabilizing nonlinear controllers for euler lagrange systems reported in the literature our objective in this study is twofold first to compare the performance of these schemes from a perspective different from stabilizability second to quantify the basic tradeoff between robust stability and robust performance for these designs we consider passivity based and feedback linearization schemes developed for the control of dc to dc converters and rigid robots for the dc to dc problem we show that for both controllers there exists a lower bound to the achievable attenuation level i e a lower bound to the l 2 gain of the closed loop operator from disturbance to regulated output which is independent of the design parameters also for the passivity based scheme we obtain an upper bound for the disturbance attenuation which is insured provided we sacrifice the convergence rate for rigid robots we show that both approaches yield arbitrarily good disturbance attenuation without compromising the convergence rate

passivity based controllers for the stabilization of dc to dc power converters abstract passivity based feedback controllers are derived for the indirect stabilization of the average output voltage in pulse width modulation pwm controlled dc to dc power converters of the boost buck boost and buck types the controller design is carried out on the basis of well known average pwm models of such circuits the average models are first shown to be euler lagrange systems corresponding to a suitable set of average euler lagrange parameters the proposed regulators are based on an energy shaping plus damping injection scheme achievable through nonlinear dynamical feedback the performance of the proposed passivity based duty ratio synthesis policies is tested for the boost converter case the regulating feedback law derived on the basis of a boost model composed of ideal switches and ideal circuit components is assessed via computer simulations on a realistic stochastically perturbed switched converter model including parasitic resistances and parasitic voltage sources

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

numerical solutions of nonlinear fractional differential equations by alternative legendre polynomials abstract in this paper numerical techniques are presented for solving initial value problems of nonlinear fractional differential equations the method is implemented by applying alternative legendre polynomials the operational matrix of fractional integration and the product for the alternative legendre polynomials are derived in order to transform the nonlinear equations into a system of algebraic equations the study of the error analysis of the obtained method is also considered furthermore numerical examples demonstrate that this method is applicable and accurate

numerical solution of integral equations system of the second kind by block pulse functions abstract this paper endeavors to formulate block pulse functions to propose solutions for the fredholm integral equations system to begin with we describe the characteristic of block pulse functions and will go on to indicate that through this method a system of fredholm integral equations can be reduced to an algebraic equation numerical examples presented to illustrate the accuracy of the method

kronecker operational matrices for fractional calculus and some applications the problems of systems identification analysis and optimal control have been recently studied using orthogonal functions the specific orthogonal functions used up to now are the walsh the block pulse the laguerre the legendre haar and many other functions in the present paper several operational matrices for integration and differentiation are studied we introduce the kronecker convolution product and expanded to the riemann liouville fractional integral of matrices for some applications it is often not necessary to compute exact solutions approximate solutions are sufficient because sometimes computational efforts rapidly increase with the size of matrix functions our method is extended to find the exact and approximate solutions of the general system matrix convolution differential equations the way exists which transform the coupled matrix differential equations into forms for which solutions may be readily computed finally several systems are solved by the new and other approaches and illustrative examples are also considered

numerical solutions of volterra integral equation with weakly singular kernel using scw method numerical methods for weakly singular volterra integral equations are rarely considered in the literature the solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration and bring some difficulties in numerical computation in this paper we present a numerical solution of weakly singular volterra integral equations including the abels equations by the second chebyshev wavelet scw we give the scw operational matrix of fractional integration and combine with the block pulse functions bpfs to derive the procedure of solving this kind integral equations the proposed method is illustrated with numerical examples the results reveal that the method is accurate and easy to implement

brief paper tracking and disturbance rejection for fully actuated mechanical systems in this paper we solve the tracking and disturbance rejection problem for fully actuated passive mechanical systems we assume that the reference signal r and its first two derivatives r r are available to the controller and the disturbance signal d can be decomposed into a finite superposition of sine waves of arbitrary but known frequencies and an arbitrary l 2 signal we combine the internal model principle with the ideas behind the slotine li adaptive controller the internal model based adaptive controller that we propose causes the closed loop state trajectories to be bounded and the tracking error and its derivative to converge to zero without any prior knowledge of the plant parameters an important part of our results is that we prove the existence and uniqueness of the state trajectories of the closed loop system

future polynomial regularization of ill posed volterra equations we examine a new discrete method for regularizing ill posed volterra problems unlike many classical regularization techniques such as tikhonov regularization this method preserves the volterra causal nature of the problem allowing the regularized solution to be produced sequentially in near real time as data arrives we analyze the method and for an important class of volterra problems prove that regularized solutions converge to the true solution at the best possible rate with respect to error in the data in fact the future polynomial regularization method discussed here may be applied to quite general operator equations provided that the operator may be discretized by a lower triangular matrix of toeplitz type this enlarges the class of operator equations that may be approximated using the method but it also introduces degenerate situations in which the future polynomial method is no more regularizing than an ordinary discretization method we characterize these degenerate cases and argue that we are unlikely to see them for the problems of interest here in particular such degeneracies cannot occur for the class of volterra problems for which we are able to prove the future polynomial method converges finally we present numerical evidence that this method works well in the recovery of sharp and discontinuous features in the true solution features that can be oversmoothed by classical regularization techniques

convergence properties of a class of product formulas for weakly singular integral equations we examine the convergence of product quadrature formulas of interpolatory type based on the zeros of certain generalized jacobi polynomials for the discretization of integrals of the type f k x y f x dx l 1 and f x has algebraic singularities of the form 1 x a a 1 we prove that the uniform rate of convergence of the rules is o m log2 m in the case of the first kernel and o m 2 a2v logim if v 0 for the second where m is the number of points in the quadrature rule

some new connections between matrix products for partitioned and non partitioned matrices this paper is concerned with a long list of matrix products for partitioned and non partitioned matrices including kronecker khatri rao of first kind hadamard tracy singh khatri rao block kronecker block hadamard and box products sums our contribution is to gather the most useful connections of the matrix products from various sources and present them in one place several new attractive connections are also added to this collection we give as an example two applications in order to show that these connections play a central role in many applications these applications are a block diagonal least squares problem and a generalization of matrix inequalities involving khatri rao products of positive definite matrices

improved algorithm about nsfot for the purpose of further saving computing time an improved algorithm about nsfot is provided in this paper that is by introducing the simple operations such as preprocessing or after processing haar and walsh transforms are performed conveniently on the multiprocessor as a result one large size problem is divided into several small size sub problems load on every processor not only decreases greatly but also gets so uniform that much time is saved both the theoretical analysis and experimental results demonstrate the effectiveness of the proposed approach

numerical solution of first kind volterra equations by sequential tikhonov regularization we consider the problem of finding regularized solutions to ill posed volterra integral equations the method we consider is a sequential form of tikhonov regularization that is particularly suited to problems of volterra type we prove that when this sequential regularization method is coupled with several standard discretizations of the integral equation collocation rectangular and midpoint quadrature one obtains convergence of the method at an optimal rate with respect to noise in the data in addition we describe a fast algorithm for the implementation of sequential tikhonov regularization and show that for small values of the regularization parameter the method is only slightly more expensive computationally than the numerical solution of the original unregularized integral equation finally numerical results are presented to show that the performance of sequential tikhonov regularization nearly matches that of standard tikhonov regularization in practice but considerable savings in cost are realized

the third kind chebyshev wavelets collocation method for solving the time fractional convection diffusion equations with variable coefficients in this paper a numerical method based on the third kind chebyshev wavelets is proposed for solving a class of time fractional convection diffusion equations with variable coefficients the third kind chebyshev wavelets operational matrices of the integer order integration and the fractional order integration are derived respectively they are utilized to reduce the problem to a system of linear algebraic equations by combining the collocation method the uniform convergence analysis and error estimation for the third kind chebyshev wavelets expansion are investigated illustrative examples are given and the numerical results are presented to demonstrate the efficiency and accuracy of the proposed method

the effect of initial stress and orthotropy on the propagation waves in a hollow cylinder in this paper the effect of the initial stress on the propagation of axisymmetric elastic waves in hollow orthotropic circular cylinder is investigated on the basis of the linear theory of elasticity the equations of elastodynamic are solved using the method of power series furthermore the frequency equation for the phase velocity of waves is obtained the results indicated that the effect of orthotropy on such waves was small while the effect of initial stress was very much pronounced

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

modified chebyshev wavelet methods for fractional delay type equations in this article we develop the chebyshev wavelet method for solving the fractional delay differential equations and integro differential equations according to the development we approximate the delay unknown functions by the chebyshev wavelets series at delay time which we call the delay chebyshev wavelet series we also proposed a technique by combining the method of steps and chebyshev wavelet method for solving fractional delay differential equations this technique converts the fractional delay differential equation on a given interval to a fractional non delay differential equation over that interval by using the function defined on previous interval and then apply the chebyshev wavelet method on the obtained fractional non delay differential equation to find the solution numerical examples will be presented to demonstrate the benefits of computing with these approaches

numerical solution of fractional differential equations using the generalized block pulse operational matrix the riemann liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration also the generalized block pulse operational matrices of differentiation are derived based on the above results we propose a way to solve the fractional differential equations the method is computationally attractive and applications are demonstrated through illustrative examples

an efficient computational method for solving fractional biharmonic equation in this paper we first introduce the fractional biharmonic equation and an orthogonal system of basis functions for the space of continuous functions on the interval 0 l generated by the shifted chebyshev polynomials moreover we propose a computational method based on the operational matrix of fractional derivatives of these basis functions for solving the fractional biharmonic equation the main characteristic behind this approach is that it reduces the problem under consideration to solving a system of algebraic equations which greatly simplifies the problem convergence of the shifted chebyshev polynomials expansion in two dimensions is investigated also the power of this manageable method is illustrated

an efficient approach for solving the riccati equation with fractional orders the present study introduces a novel and simple analytical method for the solution of fractional order riccati differential equation in this approach the solution considered as a taylor series expansion converges rapidly to the nonlinear problem new homotopy perturbation method nhpm depends only on two components of the homotopy series the method is illustrated by applications and the results obtained are compared with those of the exact solution moreover comparing the methodology with some known techniques shows that the present approach is relatively easy and efficient

superconvergence of collocation methods for a class of weakly singular volterra integral equations we discuss the application of spline collocation methods to a certain class of weakly singular volterra integral equations it will be shown that by a special choice of the collocation parameters superconvergence properties can be obtained if the exact solution satisfies certain conditions this is in contrast with the theory of collocation methods for abel type equations several numerical examples are given which illustrate the theoretical results

dynamic regularization of input estimation problems by explicit block inversion input estimation problems in structural dynamics often lead to solving damped least squares problems explicit block inversion algorithms are developed to invert the associated upper block triangular toeplitz matrix moreover the derived inversion algorithms enable the introduction of a new type of sequential regularization technique for optimal regularization parameters it is demonstrated by a numerical example that the solution using the proposed regularization technique has an upper bound equal to zeroth order tikhonov regularization

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method