Hägusad teist liiki integraalvõrrandid

Käesolevas doktoritöös on uuritud hägusaid teist liiki integraalvõrrandeid. Need võrrandid sisaldavad hägusaid funktsioone, s.t. funktsioone, mille väärtused on hägusad arvud. Me tõestasime tulemuse sileda tuumaga hägusate Volterra integraalvõrrandite lahendite sileduse kohta. Kui integraalvõrrandi tuum muudab märki, siis integraalvõrrandi lahend pole üldiselt sile. Nende võrrandite lahendamiseks me vaatlesime kollokatsioonimeetodit tükiti lineaarsete ja tükiti konstantsete funktsioonide ruumis. Kasutades lahendi sileduse tulemusi tõestasime meetodite koonduvuskiiruse. Me vaatlesime ka nõrgalt singulaarse tuumaga hägusaid Volterra integraalvõrrandeid. Uurisime lahendi olemasolu, ühesust, siledust ja hägusust. Ülesande ligikaudseks lahendamiseks kasutasime kollokatsioonimeetodit tükiti polünoomide ruumis. Tõestasime meetodite koonduvuskiiruse ning uurisime lähislahendi hägusust. Nii analüüs kui ka numbrilised eksperimendid näitavad, et gradueeritud võrke kasutades saame parema koonduvuskiiruse kui ühtlase võrgu korral. Teist liiki hägusate Fredholmi integraalvõrrandite lahendamiseks pakkusime uue lahendusmeetodi, mis põhineb kõigi võrrandis esinevate funktsioonide lähendamisel Tšebõšovi polünoomidega. Uurisime nii täpse kui ka ligikaudse lahendi olemasolu ja ühesust. Tõestasime meetodi koonduvuse ja lähislahendi hägususe.In this thesis we investigated fuzzy integral equations of the second kind. These equations contain fuzzy functions, i.e. functions whose values are fuzzy numbers. We proved a regularity result for solution of fuzzy Volterra integral equations with smooth kernels. If the kernel changes sign, then the solution is not smooth in general. We proposed collocation method with triangular and rectangular basis functions for solving these equations. Using the regularity result we estimated the order of convergence of these methods. We also investigated fuzzy Volterra integral equations with weakly singular kernels. The existence, regularity and the fuzziness of the exact solution is studied. Collocation methods on discontinuous piecewise polynomial spaces are proposed. A convergence analysis is given. The fuzziness of the approximate solution is investigated. Both the analysis and numerical methods show that graded mesh is better than uniform mesh for these problems. We proposed a new numerical method for solving fuzzy Fredholm integral equations of the second kind. This method is based on approximation of all functions involved by Chebyshev polynomials. We analyzed the existence and uniqueness of both exact and approximate fuzzy solutions. We proved the convergence and fuzziness of the approximate solution.https://www.ester.ee/record=b539569

A Hybrid Collocation Method for Volterra Integral Equations with Weakly Singular Kernels

The commonly used graded piecewise polynomial collocation method for weakly singular Volterra integral equations may cause serious round-off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time-dependent equations to round-off errors. The singularity preserving ( nonpolynomial) collocation method is known to have only local convergence. To overcome the shortcoming of these well-known methods, we introduce a hybrid collocation method for solving Volterra integral equations of the second kind with weakly singular kernels. In this hybrid method we combine a singularity preserving ( nonpolynomial) collocation method used near the singular point of the derivative of the solution and a graded piecewise polynomial collocation method used for the rest of the domain. We prove the optimal order of global convergence for this method. The convergence analysis of this method is based on a singularity expansion of the exact solution of the equations. We prove that the solutions of such equations can be decomposed into two parts, with one part being a linear combination of some known singular functions which reflect the singularity of the solutions and the other part being a smooth function. A numerical example is presented to demonstrate the effectiveness of the proposed method and to compare it to the graded collocation method

Keskosa interpolatsioonil põhinevad meetodid nõrgalt singulaarsete integraalvõrrandite lahendamiseks

Paljud keemia, polümeeride füüsika, matemaatilise füüsika jt teadusalade probleemid on formuleeritavad integraalvõrrandite kujul ning nende probleemide käsitlus taandub integraalvõrrandite lahendamisele või kvalitatiivsele uurimisele. Integraalvõrrandeid, mida saab täpselt lahendada, on suhteliselt vähe, seega on väga olulised meetodid võrrandite numbriliseks lahendamiseks. Käesolevas doktoritöös pakume välja kaks kõrget järku numbrilist meetodit, mis ei kasuta lineaarse teist liiki singulaarsustega Fredholmi integraalvõrrandi lahendamiseks ebaühtlast võrku. Need meetodid on kollokatsioonimeetod ja korrutise integreerimise meetod. Nimetatud meetodid põhinevad keskosa interpolatsioonil polünoomidega ühtlasel võrgul ja silendaval muutujate vahetusel. Lõigu keskosas on interpolatsioonivea hinnang ligikaudu 2m korda täpsem kui kogu lõigul. Lisaks on interpolatsiooniprotsess ühtlasel võrgul lõigu keskosas m-i kasvades stabiilne. Muutujate vahetuse abil parendame me võrrandi täpse lahendi käitumist. Doktoritöös on kirjeldatud toodud meetodite koondumist ja koondumiskiirustThere are a number of problems from many different fields, for example chemistry, physics of polymers and mathematical physics, which are directly formulated in terms of integral equations; and there are problems that are represented in terms of differential equations with auxiliary conditions, but which can be reduced to integral equations. There are relatively few integral equations which can be solved exactly, hence, numerical schemes are required for dealing with these equations in a proper manner. In this thesis we propose two new classes of high order numerical methods, which do not need graded grids for solving linear Fredholm integral equations of the second kind with singularities. The methods are developed by means of the 'central part' interpolation by polynomials on the uniform grid and smoothing change of variables. In the central parts of the interval, the estimates of interpolation error are approximately 2m times more precise than on the whole interval. In the central parts of the interval, the interpolation process on the uniform grid also has good stability properties as m increases. With the help of a change of variables we improve the boundary behaviour of the exact solution of the problem. The convergence and the convergence order of methods is studied

An improvement of the product integration method for a weakly singular Hammerstein equation

We present a new method to solve nonlinear Hammerstein equations with weakly singular kernels. The process to approximate the solution, followed usually, consists in adapting the discretization scheme from the linear case in order to obtain a nonlinear system in a finite dimensional space and solve it by any linearization method. In this paper, we propose to first linearize, via Newton method, the nonlinear operator equation and only then to discretize the obtained linear equations by the product integration method. We prove that the iterates, issued from our method, tends to the exact solution of the nonlinear Hammerstein equation when the number of Newton iterations tends to infinity, whatever the discretization parameter can be. This is not the case when the discretization is done first: in this case, the accuracy of the approximation is limited by the mesh size discretization. A Numerical example is given to confirm the theorical result

Numerical Methods for Integral Equations

We first propose a multiscale Galerkin method for solving the Volterra integral equations of the second kind with a weakly singular kernel. Due to the special structure of Volterra integral equations and the ``shrinking support property of multiscale basis functions, a large number of entries of the coefficient matrix appearing in the resulting discrete linear system are zeros. This result, combined with a truncation scheme of the coefficient matrix, leads to a fast numerical solution of the integral equation. A quadrature method is designed especially for the weakly singular kernel involved inside the integral operator to compute the nonzero entries of the compressed matrix so that the quadrature errors will not ruin the overall convergence order of the approximate solution of the integral equation. We estimate the computational cost of this numerical method and its approximate accuracy. Numerical experiments are presented to demonstrate the performance of the proposed method. We also exploit two methods based on neural network models and the collocation method in solving the linear Fredholm integral equations of the second kind. For the first neural network (NN) model, we cast the problem of solving an integral equation as a data fitting problem on a finite set, which gives rise to an optimization problem. In the second method, which is referred to as the NN-Collocation model, we first choose the polynomial space as the projection space of the Collocation method, then approximate the solution of the integral equation by a linear combination of polynomials in that space. The coefficients of the linear combination are served as the weights between the hidden layer and the output layer of the neural network. We train both neural network models using gradient descent with Adam optimizer. Finally, we compare the performances of the two methods and find that the NN-Collocation model offers a more stable, accurate, and efficient solution

On the stability of piecewise polynomial collocation methods for solving weakly singular integral equations of the second kind

Piecewise polynomial collocation methods on special nonuniform grids are efficient methods for solving weakly singular Fredholm and Volterra integral equations but there is a widespread belief that those methods are numerically unstable in the case of large values of the nonuniformity parameter r. We show that this method by itself is stable and discuss some implementation problems that may lead to unstable behavior of numerical results. First Published Online: 14 Oct 201