Solution Of Integral Equation Research Articles

Item identification with a space-dependent model of neutron multiplicities and artificial neural networks

A method of calculating the neutron multiplicity rates (singles, doubles and triples rates), based on transport theory, was developed by us recently. The model treats the full 3-D spatial transport and multiplication of neutrons, accounting also for the shape of the item and the spatial distribution of the source, in one-speed theory. For a given item and its source distribution, the model can predict the multiplicity rates more precisely than the point model, on which traditional neutron multiplicity counting is based. However, so far it has not been investigated how the enhanced accuracy of the calculated multiplicity rates (i.e. the solution of the direct task) can be used to estimate the parameters of interest of the measurement item, primarily the fission rate (the solution of the inverse task). Unlike for the point model, the multiplicity rates under the extended scheme can only be given numerically, as solutions of integral transport equations, and hence an analytical inversion of the formulae is not possible. In this work it is investigated how machine learning methods, primarily the use of artificial neural networks, which only need numerical values of the solution of the direct task (the multiplicity rates), can be used for this purpose. It is shown that for numerical test items containing a mixture of 239Pu and 240Pu, the fraction of the latter varying between 4% and 25%, one can extract the masses of both isotopes from a properly trained network.

Reich–Krasnoselskii-type fixed point results with applications in integral equations

In this paper, motivated by Reich contraction and tool of measure of noncompactness, some generalizations of Reich, Kannan, Darbo, Sadovskii, and Krasnoselskii type fixed point results are presented by considering a pair of maps A, B on a nonempty closed subset M of a Banach space X into X. The existence of a solution to the equation Ax+Bx=x, where A is k-set contractive and B is a generalized Reich contraction, is established. As applications, it is established that the main result of this paper can be applied to learn conditions under which a solution of a nonlinear integral equation exists. Further we explain this phenomenon with the help of a practical example to approximate such solutions by using fixed point techniques. The graphs of exact and approximate solutions are also given to attract readers for further research activities.

On the uniqueness of continuous positive solution for a non-linear integral equation whose singularity lies in the reciprocal of the solution

On the uniqueness of continuous positive solution for a non-linear integral equation whose singularity lies in the reciprocal of the solution

Reactor physics of research reactors at BARC: Aspects and scopes

Research reactors have a wide range of applications in the fields like medicine, agriculture, industry, development and validation of reactor physics codes, condensed matter research, etc. Reactor physics plays very important role in the design and operation of nuclear reactors. The nuclear data, the methods of solution of neutron integral and integro-differential transport equation, diffusion equation, both steady and time dependent have undergoes tremendous improvement over last few decades due to availability of fast computing machines with large memory. The whole analysis of a reactor requires developments of lattice code for lattice level calculations, multigroup diffusion codes for core calculations and point kinetics & space time kinetics codes for reactivity-initiated transients, on-line reactivity measurements using inverse kinetics method, a point depletion code for estimating radioactivity or concentration of fission products/actinides, etc. Reactor physics also plays key role in developing the neutron gamma coupled shielding codes. The reactor safety, which requires the coupling of kinetics equations, heat conduction equations, coolant dynamics equations and hydrodynamics equations is also very important especially after Fukushima accidents. BARC has three functional research reactors currently. In this paper, we discuss important reactor physics aspects and the scope of development with respect to research reactors at Trombay.

On the calculation of functionals from the solution of the linear Skorohod SDE with first-order chaos in the coefficients

This article is devoted to the precise and approximate calculation of the mathematical expectation of non-linear functionals from the solution of the linear Skorohod equation with first-order chaos in the coefficients and the initial condition. In [1–4], approximate methods for calculating the mathematical expectation of functionals from solutions of the linear Skorohod stochastic differential equation with a random initial condition and deterministic coefficient functions were proposed and investigated. This paper considers the calculation of the mathematical expectation of nonlinear functionals from the solution to the linear Skorohod equation with first-order chaos in the coefficients and the initial condition. In this case, the solution is obtained in an analytical form [5]; however, it contains an unknown random parameter, determined as the solution of an auxiliary integral stochastic equation. In this paper we investigate the cases when the solution of this integral equation is found in an explicit form and then evaluate the moments and the mathematical expectations of some types of functional from the solution of the initial Skorohod equation. The construction of approximate formulas for calculating more general nonlinear functionals from the solution is considered. Numerical examples are given to illustrate the accuracy of the obtained formulas.

Existence, uniqueness, and numerical approximation of solutions of a nonlinear functional integral equation

A general nonlinear functional integral equation has been studied. The considered equation is of Fredholm type and includes some well-known integral or functional equations, as special cases. By assuming some smoothness conditions on the functions in the equation, we prove the existence and uniqueness of the solution by means of a fixed point method. Then, we apply the so-called Nyström method in order to approximate the solution, which leads to a system of nonlinear algebraic equations. In the next step, the Picard iteration method is utilized for approximating the solution of the resulting system of nonlinear algebraic equations. The convergence of the numerical scheme is proved, its computational complexity is computed and finally some illustrative examples are included to demonstrate the efficiency and applicability of the technique.

On solutions of differential and integral equations using new fixed point results in cone [formula omitted]-metric spaces

The focus of this study is to establish the existence and uniqueness of solutions for differential and integral equations within specific metric spaces. Our investigation begins by introducing the concept of the so-called cone Eb-metric space and presenting crucial findings in this particular space. We have presented fixed point results for specific contractions, particularly in the context of non-solid cones that possess semi-interior points. Not only do the results enhance specific previous fixed points outcomes, but they also encompass and extend previous findings documented in the literature. Furthermore, we apply our findings in the cone Eb-metric space to various examples and applications. The ultimate outcome is the rigorous validation of the existence and uniqueness of solutions for certain differential and integral equations.

Numerical solution of nonlinear third-kind Volterra integral equations using an iterative collocation method

In this paper, we discuss the application of an iterative collocation method based on the use of Lagrange polynomials for the numerical solution of a class of nonlinear third-kind Volterra integral equations. The approximate solution is given by explicit formulas. The error analysis of the proposed numerical method is studied theoretically. Some numerical examples are given to confirm our theoretical results.

Existence and Uniqueness Solution of Certain Integral Equation

The aim of this work is to study the existence and uniqueness of solutions of certain integral equations by using the Picard approximation method and Banach fixed point theorem. The study of integral equations is more general and leads us to improve and extend some results of Butris.

Multi-level Power Series Solution for Large Surface and Volume Electric Field Integral Equation

In this paper, we propose a new multi-level power series solution method for solving a large surface and volume electric field integral equation-based H-Matrix. The proposed solution method converges in a fixed number of iterations and is solved at each level of the H-Matrix computation. The solution method avoids the computation of a full matrix, as it can be solved independently at each level, starting from the leaf level. Solution at each level can be used as the final solution, thus saving the matrix computation time for full H-Matrix. The paper shows that the leaf level matrix computation and solution with power series gives as accurate results as the full H-Matrix iterative solver method. The method results in considerable savings time and memory savings compared to the H-Matrix iterative solver. Further, the proposed method retains the O(NlogN) solution complexity.

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

We have four kinds of solutions for Cauchy type integral equations: by expanding on known functions and using the Maclaurin series, we can convert these four kinds of solutions into linear combinations of some elements that are basis of these solutions. Using these bases gives exact solutions for polynomials and, for some other functions, a high-accuracy approximate solution.

An iterative Nyström-based method to solve nonlinear Fredholm integral equations of the second kind

In the current paper, an iterative numerical scheme based upon the quasilinearization technique and Nyström method to find the approximate solution of the nonlinear Fredholm integral equations of the second kind is provided. First, using the quasilinearization method, the nonlinear Fredholm integral equation is reduced to a sequence of linear Fredholm integral equations. Under some suitable assumptions, the exact solutions of this sequence of linear integral equations converge to the unique solution of the original problem quadratically. Then in each iteration, utilizing the linear barycentric rational quadrature, the linear Fredholm integral equation is approximated with the Nyström method. Convergence analysis of the method is investigated thoroughly. Numerical examples and comparisons with other existing methods are provided to confirm the theoretical results and demonstrate the performance and validity of the numerical method.

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

Constructing Solutions of Cauchy Type Integral Equations by Using Four Kinds of Basis

Existence of solution of Erdelyi-Kober fractional integral equations using measure of non-compactness

Existence of solution of Erdelyi-Kober fractional integral equations using measure of non-compactness

Approximate Analytical Solution of Fuzzy Linear Volterra Integral Equation via Elzaki ADM

In this paper, the fuzzy Volterra integral equations’ solutions are calculated using a hybrid methodology. The combination of the Elzaki transform and Adomian decomposition method results in the development of a novel regime. The precise fuzzy solutions are determined using Elzaki ADM after the fuzzy linear Volterra integral equations are first translated into two crisp integral equations utilizing the fuzzy number in parametric form. Three instances of the considered equations are solved to show the established scheme’s dependability, efficacy, and application. The results have a substantial impact on the fuzzy analytical dynamic equation theory. The comparison of the data in a graphical and tabular format demonstrates the robustness of the defined regime. The lower and upper bound solutions’ theoretical convergence and error estimates are highlighted in this paper. A tolerable order of absolute error is also obtained for this inquiry, and the consistency of the outcomes that are approximated and accurate is examined. The regime generated effective and reliable results. The current regime effectively lowers the computational cost, and a faster convergence of the series solution to the exact answer is signaled.

Finite temperature properties of an integrable zigzag ladder chain

We consider the interaction-round-a-face version of the six-vertex model for arbitrary anisotropy parameter, which allow us to derive an integrable one-dimensional quantum Hamiltonian with three-spin interactions. We apply the quantum transfer matrix approach for the face model version of the six-vertex model. The integrable quantum Hamiltonian shares some thermodynamical properties with the Heisenberg XXZ chain, but has different ordering and critical exponents. Two gapped phases are the dimerized antiferromagnetic order and the usual antiferromagnetic (Néel) order for positive nearest neighbor Ising coupling. In between these, there is an extended critical region, which is a quantum spin-liquid with broken parity symmetry inducing an oscillatory behavior at the long distance <σ1zσℓ+1z> correlation. At finite temperatures, the numerical solution of the non-linear integral equations allows for the determination of the correlation length as well as for the momentum of the oscillation.

Numerical solution of stochastic Volterra integral equations based on uniform Haar wavelets by using direct method

Nowadays, wavelets have played a crucial role in approximating the solution of a wide range of problems arising in science and engineering. Wavelets have been used in numerous areas of applied mathematics as diverse as signal analysis, statistics, computer-aided geometric, image processing and numerical analysis. The uniform Haar wavelets are most widely used wavelets for solving stochastic integral equations, outperforming Legendre, Daubechies, etc. The uniform Haar wavelets are generated from pairs of piecewise constant functions and can be simply integrated. Furthermore, the uniform Haar functions are orthogonal and form a good transform basis. In this paper, we have proposed an efficient numerical method to solve a class of stochastic linear Volterra integral equations (SLVIEs) based on the uniform Haar wavelets. The aim of this paper is to apply numerical method to obtain approximate solution of SLVIEs. By using this method, the SLVIEs can be reduced to a linear system of algebraic equations. Also, the error analysis of this method is performed. Finally, some numerical examples are given to demonstrate pertinent features of this method. Furthermore, we compare obtained results from this method with the achieved results from relevant studies.

A numerical investigation of crack behavior near a fixed boundary using singular integral equation and finite element methods

In this paper, we present the method of integral transforms and the Gauss-Chebyshev quadrature methods to solve the problem of a crack parallel to a fixed boundary under remote tension. We derive a system of singular integral equations of the first kind, specific to the problem at hand, which we numerically solve using Gauss-Chebyshev integration. We specialize our results to the problem of a crack in an infinite plate under remote tension, and show that the relative error in our numerically derived solutions is within machine precision of the closed-form analytical solutions. Stress intensity factors are calculated that are in excellent agreement with those derived by others using different methods. We also demonstrate that both the stress intensity factors and normal σyy(x,y) and shear σxy(x,y) stress fields derived via numerical solution of the singular integral equations, compare well with those determined using the commercially available Abaqus finite element code where the crack is modeled using the eXtended Finite Element Method.

New fixed point theorems via measure of noncompactness and its application on fractional integral equation involving an operator with iterative relations

In this paper, Darbo’s fixed point theorem is generalized and it is applied to find the existence of solution of a fractional integral equation involving an operator with iterative relations in a Banach space. Moreover, an example is provided to illustrate the results.

A Class of Relational Functional Contractions with Applications to Nonlinear Integral Equations

In this article, we investigate some fixed-point results under certain functional contractive mappings in a relation metric space. In the process, we utilize more general contraction condition which must be verified for comparative elements only. Our results enrich, modify, refine, unify and sharpen several existing fixed-point results. We construct some examples in support of our results. To attest to the applicability of our results, we establish the existence and uniqueness of theorems regarding the solutions of certain nonlinear integral equations.