Multidimensional Integral Equations Research Articles

Nonlocal symmetries and solutions of the multi-dimensional integrable long water wave equations

In this paper, the (2+1)-dimensional integrable long water wave equations (LWWs) are constructed for the first time using the conservation law of the (1+1)-dimensional LWWs. The new (1+1)-dimensional LWWs can be obtained by introducing a constraint to the (2+1)-dimensional LWWs. This new (1+1)-dimensional LWWs are studied by using nonlocal symmetry methods for the first time. The closed system corresponding to nonlocal symmetry is established by the lax pairs of equations and the potential function determined using conservation laws. Exact solutions of the equations are obtained by finite symmetry transformation and symmetry approximation of this closed system. The dynamic behavior of the equations is studied by means of figures of the exact solutions.

A THEORY OF MULTIDIMENSIONAL FREDHOLM INTEGRAL EQUATIONS HAVING SEPARABLE KERNELS: SOLVABLE IN A REGION SURROUNDING BY THE HYPERPLANES

In this article, we present a theory of multidimensional Fredholm integral equations, having separable kernels, are solvable in a region surrounded by hyperplanes. In derivation of their solutions, we employ the generalized Hilbert-Schmidt theory involving eigenvalues and corresponding normalized eigen functions obtained by separable kernels in a region surrounded by the hyperplanes. Finally, we apply two variables Gegenbauer polynomials and derive a result on the inequality of the solution for the double symmetric Fredholm integral equation.

Advanced Stochastic Approaches for Fredholm Integral Equations

Integral equations are of high applicability in different areas of applied mathematics, physics, engineering, geophysics, electricity and magnetism, kinetic theory of gases, quantum mechanics, mathematical economics, and queuing theory. That is why it is reasonable to develop and study efficient and reliable approaches to solve integral equations. For multidimensional problems the existing biased stochastic algorithms based on evaluation of finite number of integrals will suer more from the effect of high dimensionality, because they are based on quadrature points. So we need advanced unbiased algorithms for solving the multidimensional problem which is developed in this paper. A new unbiased stochastic method for solving multidimensional Fredholm integral equations of second kind is proposed and analysed. We compared the newly proposed unbiased algorithm with the old unbiased stochastic algorithm for the one dimensional problem and multidimensional problem.

Using Homotopy Perturbation and Analysis Methods for Solving Different-dimensions Fractional Analytical Equations

The aim of the research, we extended the one-dimensional to multi-dimensional, we applied the homotopy perturbation and analysis methods to solve Volterra integral equations and to obtain approximate analytical solutions of systems of the second kind multi-dimensional Volterra integral equations. We proved the convergence of the homotopy analysis method (HAM). The HAM solutions contained an auxiliary parameter that provides a convenient way of controlling the convergence region of series solutions. It is shown that the solutions obtained by the homotopy-perturbation method (HPM) are only special cases of the HAM solutions. Several examples are given to illustrate the efficiency and implementation of the method. The results indicate that this method is efficient for the linear and non - linear models with the dissipative terms.

New Aspects on the Solvability of a Multidimensional Functional Integral Equation with Multivalued Feedback Control

The current study demonstrates the existence of solutions to a multidimensional functional integral equation with multivalued feedback. We seek solutions for the multidimensional functional problem that is defined, continuous, and bounded on the semi-infinite interval. Our proof is based on the technique associated with measures of noncompactness by a given modulus of continuity in the space in BC(R+). Also, some sufficient conditions are investigated to demonstrate the asymptotic stability of the solutions to that multidimensional functional equation. Additionally, we give an example and some particular cases to illustrate our outcomes.

A combination of the quasilinearization method and linear barycentric rational interpolation to solve nonlinear multi-dimensional Volterra integral equations

In this paper, an iterative scheme including a combination of the quasilinearization technique and multi-dimensional linear barycentric rational interpolation is applied to solve nonlinear multi-dimensional Volterra integral equations. First, employing the quasilinearization method, the nonlinear multi-dimensional Volterra integral equation is reduced to a sequence of linear Volterra integral equations. Under appropriate assumptions, the constructed iterative sequence is uniformly convergent to the unique solution of the problem. In general, finding an analytical solution for linear Volterra integral equations is impossible. Hence, in each iteration, using a collocation method based on multi-dimensional barycentric rational basis functions, the solution to the linear integral equation is approximated. The quadratic convergence of the quasilinearization approach and the error estimation of the combined method are investigated theoretically. In the end, the efficiency and the validity of this method are illustrated with some numerical examples and compared with those of the existing numerical methods.

Methods and effective algorithms for solving multidimensional integral equations

Objectives. Integral equations have long been used in mathematical physics to demonstrate existence and uniqueness theorems for solving boundary value problems for differential equations. However, despite integral equations have a number of advantages in comparison with corresponding boundary value problems where boundary conditions are present in the kernels of equations, they are rarely used for obtaining numerical solutions of problems due to the presence of equations with dense matrices that arise that when discretizing integral equations, as opposed to sparse matrices in the case of differential equations. Recently, due to the development of computer technology and methods of computational mathematics, integral equations have been used for the numerical solution of specific problems. In the present work, two methods for numerical solution of two-dimensional and three-dimensional integral equations are proposed for describing several significant classes of problems in mathematical physics.Methods. The method of collocation on non-uniform and uniform grids is used to discretize integral equations. To obtain a numerical solution of the resulting systems of linear algebraic equations (SLAEs), iterative methods are used. In the case of a uniform grid, an efficient method for multiplying the SLAE matrix by vector is created.Results. Corresponding SLAEs describing the considered classes of problems are set up. Efficient solution algorithms using fast Fourier transforms are proposed for solving systems of equations obtained using a uniform grid.Conclusions. While SLAEs using a non-uniform grid can be used to describe complex domain configurations, there are significant constraints on the dimensionality of described systems. When using a uniform grid, the dimensionality of SLAEs can be several orders of magnitude higher; however, in this case, it may be difficult to describe the complex configuration of the domain. Selection of the particular method depends on the specific problem and available computational resources. Thus, SLAEs on a non-uniform grid may be preferable for many two-dimensional problems, while systems on a uniform grid may be preferable for three-dimensional problems.

A spectral collocation method for a nonlinear multidimensional Volterra integral equation

AbstractIn this paper, a spectral method is implemented to solve a nonlinear multidimensional Volterra integral equation. Under some condition, the spectral convergence analysis of the solution is investigated in both norm and norm. The results of a numerical example confirm the theoretical prediction.

Solving magnetic induction heating problem with multidimensional Fredholm integral equation methods: Alternative approach for optimization and evaluation of the process performance

Induction heating is a frequently used technology in both fundamental and applied research. It is heavily exploited in the industry for processing materials by heat treatments. In addition, it is viewed as a promising tool in medicine, particularly as a part of therapeutic strategies for treating cancer diseases. Thus, in order to optimize (i.e., enhance and tune) the performance of the induction heating process, several aspects must be considered, including the design of the magnetic coils, features of the magnetic fields applied, coupling of magnetic and thermal fields, and the material’s characteristics. To tackle this complex problem, numerical mathematical models are often used. The results of which can help in understanding the role of the various parameters on the performance of the induction heating. Here, we present an alternative mathematical approach to solve the induction heating problem using Fredholm integral equations of the second kind with a singular kernel. To reduce the computation time, the Nyström method has been adopted. As the kernel function shows a singularity, a singularity subtraction has been involved in the developed mathematical procedure. Furthermore, the error features of the Nyström method with the singularity subtraction have been described, and convergence conditions of the proposed computational algorithm have been thoroughly identified. Although special conditions for the kernel function and the integration rule are needed, the method shows lower computing times, competing well with those of traditional finite-element based routines. The applicability of the developed methodology is demonstrated for the simulation of induction heating the body of a metal object.

On One Class of Multidimensional Integral Equations of Convolution Type with Convex Nonlinearity

On One Class of Multidimensional Integral Equations of Convolution Type with Convex Nonlinearity

Гладкостные свойства оператора типа потенциала Рисса с логарифмической характеристикой

Marcel Riesz kernels generalize the ones of classical theory, and convolution with them implements the negative frac-tional powers of the Laplace operator. Along with hypersingular integrals, the Riesz potential type operators arise in new areas of analysis and its applications, for example, in various problems of mathematical physics. In the study of such problems, a significant role is played by the conditions of solvability of the corresponding multidimensional integral equations in certain function spaces, often non-classical but postulating the required analytical properties of solutions. In the presented paper, the Hardy-Littlewood type theorem is proved, considering the boundedness con-ditions for the potential with a power-logarithmic kernel and a density, integrable by Lebesgue with a specific power weight. It is shown that for the higher orders of potential, the image of a function from this class belongs to the weighted generalized Hölder space with a power-logarithmic characteristic. The action of this operator in the gener-alized Hölder spaces with a weight from the scale of power functions is also considered, including the isomorphisms of these spaces in special cases. Stereographic projection and theorems proved for the spherical Riesz potential type operators are applied. Consequently, the solvability conditions of a Poisson-type equation with a negative fractional power of the Laplace operator are obtained.

Asymptotic Behavior of Solutions of Integral Equations with Homogeneous Kernels

The multidimensional integral equation of second kind with a homogeneous of degree (−n) kernel is considered. The special class of continuous functions with a given asymptotic behavior in the neighborhood of zero is defined. It is proved that, if the free term of the integral equation belongs to this class and the equation itself is solvable, then its solution also belongs to this class. To solve this problem, a special research technique is used. The above-mentioned technique is based on the decomposition of both the solution and the free term in spherical harmonics.

Up and down deconvolution in complex geological scenarios

The redatuming approach, often referred to as up-down deconvolution, is well-known and is applied to remove water-layer and source-signature effects in seabed seismic surveys. The upgoing wavefield can be expressed as the multidimensional convolution of the downgoing wavefield with the earth’s reflectivity. Consequently, deconvolving the downgoing wavefield from the upgoing wavefield gives us the earth’s reflectivity response. The deconvolution process requires solving a multidimensional integral equation but, in a laterally invariant medium, after that wavefields are decomposed into plane-wave components, deconvolution can be enormously simplified if performed as a spectral division in the Fourier or Radon domain. It has been experimentally observed that deconvolution carried out one plane-wave component at a time gives good results, even in the presence of complex subsurface structures, provided that the seabed is relatively flat. When this geological condition is not satisfied, the same problem can be formulated in terms of interferometric redatuming using multidimensional deconvolution (MDD), in which the integral equation solution is achieved by introducing the point-spread function concept. We have developed a methodology based on numerical simulations to determine when the integral equations associated with the problem of up-down deconvolution can be solved under the assumption of shift-invariant wavefields and when it requires MDD. In the latter case, we have developed a regularized inverse procedure that mitigates the numerical problems due to the typically ill-posed nature of the inversion and that, combined with an interpolation strategy for the downgoing, enables the application of MDD within the range of sampling scenarios considered so far. We apply this methodology to synthetic data, and we discuss the potential to extend up-down deconvolution to a broader range of geological conditions.

On generalized solvability of variable nonlinear integral equations on cones

We study the solvability of multidimensional nonlinear integral equations whose operator is a sum of the Nemytsky and Urysohn type operators, and the desired solution belongs to a cone in a space of summable functions. For a wide class of nonlocal variations of the functions defining the composing operators, we prove that the equation remains solvable in the sense that its residual is arbitrarily small with respect to a suitable norm or a seminorm. The classes of acceptable variations are described by convex cones in corresponding function spaces. We also prove that images of variable operators possess the stability property in the sense that their closures under any of these variations contain a fixed convex cone.

Cubature rules based on a bivariate degree-graded alternative orthogonal basis and their applications

The purpose of this paper is to derive cubature formulas using bivariate degree-graded alternative shifted Jacobi polynomials and present some of their applications through a method for fast and explicit construction of operational integration matrices in this polynomial basis. First, we take the non-degree-graded alternative shifted Jacobi polynomial basis over the interval [0,1] and introduce a corresponding degree-graded polynomial basis. We then construct sparse and well-structured operational integration matrices in that basis. The main advantage of the proposed idea is the low computational complexity which is derived through vector–matrix multiplications without change of bases. As an application, a fast and accurate numerical algorithm based on the obtained cubature formula is developed for the approximate solution of nonlinear multi-dimensional integral equations which arise in the theory of nonlinear parabolic boundary value problems as well as the mathematical modeling of the spatio-temporal development of an epidemic. It is shown that such a matrix representation of cubature formula in terms of the bivariate degree-graded basis gives an approximate solution with a higher order accuracy. The experimental results also illustrate that smaller size and lower orders of the operational matrix and basis functions can obtain a favorable approximate solution.

Approximate Solution of Delay Integral Equations via Functions of two Variable

In this work, we adopt B. spline function of two variables basis for solving linear multi-dimensional delay Volterra integral equations nonhomogeneous of the second type. We employ the two methods to approximate solution via B. spline function of two variables basis that yields linear system. Some examples are given, the results shown in tables and figures. These methods are very effective, convenient and overcome the difficulty of traditional methods. We solve this problem with the assistance of Matlab18.

On Solvability of a Class of Multidimensional Integral Equations in the Mathematical Theory of Geographic Distribution of an Epidemic

On Solvability of a Class of Multidimensional Integral Equations in the Mathematical Theory of Geographic Distribution of an Epidemic

МУЛЬТИЛИНЕЙНЫЕ РЕШЕНИЯ ДЛЯ ДВУМЕРНОГО НЕЛИНЕЙНОГО УРАВНЕНИЯ ХИРОТЫ

At present, the question of studying multidimensional nonlinear integrable equations in the framework of the theory of solitons is very interesting to foreign and Kazakh scientists. Many physical phenomena that occur in nature can be described by nonlinearly integrated equations. Finding specific solutions to such equations plays an important role in studying the dynamics of phenomena occurring in various scientific and engineering fields, such as solid state physics, fluid mechanics, plasma physics and nonlinear optics. There are several methods for obtaining real and soliton, soliton-like solutions of such equations: the inverse scattering method, the Hirota’s bilinear method, Darboux transformation methods, the tanh-coth and the sine-cosine methods. In our work, we studied the two-dimensional Hirota equation, which is a modified nonlinear Schrödinger equation. The nonlinear Hirota equation is one of the integrating equations and the Hirota system is used in the field of study of optical fiber systems, physics, telecommunications and other engineering fields to describe many nonlinear phenomena. To date, the first, second, and n-order Darboux transformations have been developed for the two- dimensional system of Hirota equations, and the soliton, rogue wave solutions have been determined by various methods. In this article, we consider the two-dimensional nonlinear Hirota equations. Using the Lax pair and Darboux transformation we obtained the first and the second multi-line soliton solutions for this equation and provided graphical representation.

On the rate of convergence of the Legendre spectral collocation method for multi-dimensional nonlinear Volterra–Fredholm integral equations

While the approximate solutions of one-dimensional nonlinear Volterra–Fredholm integral equations with smooth kernels are now well understood, no systematic studies of the numerical solutions of their multi-dimensional counterparts exist. In this paper, we provide an efficient numerical approach for the multi-dimensional nonlinear Volterra–Fredholm integral equations based on the multi-variate Legendre-collocation approach. Spectral collocation methods for multi-dimensional nonlinear integral equations are known to cause major difficulties from a convergence analysis point of view. Consequently, rigorous error estimates are provided in the weighted Sobolev space showing the exponential decay of the numerical errors. The existence and uniqueness of the numerical solution are established. Numerical experiments are provided to support the theoretical convergence analysis. The results indicate that our spectral collocation method is more flexible with better accuracy than the existing ones.

Приближенное решение многомерных интегральных уравнений методами Монте Карло и квази-Монте Карло

The article considers an approach based on the random cubature method for solving both single and multidimensional singular integral equations, Volterra and Fredholm equations of the 1st kind, for ill-posed problems in the theory of integral equations, etc. A variant of the quasi-Monte Carlo method is studied. The integral in an integral equation is approximated using the traditional Monte Carlo method for calculating integrals. Multidimensional interpolation is applied on an arbitrary set of points. Examples of applying the method to a one-dimensional integral equation with a smooth kernel using both random and low-dispersed pseudo-random nodes are considered. A multidimensional linear integral equation with a polynomial kernel and a multidimensional nonlinear problem – the Hammerstein integral equation – are solved using the Newton method. The existence of several solutions is shown. Multidimensional integral equations of the first kind and their solution using regularization are considered. The Monte Carlo and quasi-Monte Carlo methods have not been used to solve such problems in the studied literature. The Lavrentiev regularization method was used, as well as random and pseudo-random nodes obtained using the Halton sequence. The problem of eigenvalues is solved. It is established that one of the best methods considered is the Leverrier-Faddeev method. The results of solving the problem for a different number of quadrature nodes are presented in the table. An approach based on parametric regularization of the core, an interpolation-projection method, and averaged adaptive densities are studied. The considered methods can be successfully applied in solving spatial boundary value problems for areas of complex shape. These approaches allow us to expand the range of problems in the theory of integral equations solved by Monte Carlo and quasi-Monte Carlo methods, since there are no restrictions on the value of the norm of the integral operator. A series of examples demonstrating the effectiveness of the method under study is considered.