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Journal articles on the topic 'Integral Equation Approach'

Author: Grafiati

Published: 6 September 2023

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1

Haslinger, Jaroslav, C. C. Baniotopoulos, and Panagiotis D. Panagiotopoulos. "A boundary multivalued integral “equation” approach to the semipermeability problem." Applications of Mathematics 38, no. 1 (1993): 39–60. http://dx.doi.org/10.21136/am.1993.104533.

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2

Şenel, Ayşe Anapalı, Yalçın Öztürk, and Mustafa Gülsu. "New Numerical Approach for Solving Abel’s Integral Equations." Foundations of Computing and Decision Sciences 46, no. 3 (September 1, 2021): 255–71. http://dx.doi.org/10.2478/fcds-2021-0017.

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Abstract In this article, we present an efficient method for solving Abel’s integral equations. This important equation is consisting of an integral equation that is modeling many problems in literature. Our proposed method is based on first taking the truncated Taylor expansions of the solution function and fractional derivatives, then substituting their matrix forms into the equation. The main character behind this technique’s approach is that it reduces such problems to solving a system of algebraic equations, thus greatly simplifying the problem. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. Figures and tables are demonstrated to solutions impress. Also, all numerical examples are solved with the aid of Maple.

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3

WHITFIELD, A. H., and N. MESSALI. "Integral-equation approach to system identification." International Journal of Control 45, no. 4 (April 1987): 1431–45. http://dx.doi.org/10.1080/00207178708933819.

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4

KNOBLES, D. P., S. A. STOTTS, R. A. KOCH, and T. UDAGAWA. "INTEGRAL EQUATION COUPLED MODE APPROACH APPLIED TO INTERNAL WAVE PROBLEMS." Journal of Computational Acoustics 09, no. 01 (March 2001): 149–67. http://dx.doi.org/10.1142/s0218396x01000449.

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A two-way coupled mode approach based on an integral equation formalism is applied to sound propagation through internal wave fields defined at the 1999 Shallow Water Acoustics Modeling Workshop. Solutions of the coupled equations are obtained using a powerful approach originally introduced in nuclear theory and also used to solve simple nonseparable problems in underwater acoustics. The basic integral equations are slightly modified to permit a Lanczos expansion to form a solution. The solution of the original set of integral equations is then easily recovered from the solution of the modified equations. Two important aspects of the integral equation method are revealed. First, the Lanczos expansion converges faster than a Born expansion of the original integral equations. Second, even when the Born expansion diverges due to strong mode coupling, the Lanczos expansion converges. It is shown that the internal wave problems examined are essentially one-way propagation problems because one observes good agreement between the coupled mode solutions and those provided by an energy-conserving parabolic equation algorithm. In the Workshop examples, at both 25 and 250 Hz, significantly greater coupling between modes occurs in the linear internal wave field case than the nonlinear soliton case.

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5

Wei, Tao, and Mingtian Xu. "An integral equation approach to the unsteady convection–diffusion equations." Applied Mathematics and Computation 274 (February 2016): 55–64. http://dx.doi.org/10.1016/j.amc.2015.10.084.

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6

Zozulya, V. V. "Divergent Integrals in Elastostatics: General Considerations." ISRN Applied Mathematics 2011 (August 2, 2011): 1–25. http://dx.doi.org/10.5402/2011/726402.

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This article considers weakly singular, singular, and hypersingular integrals, which arise when the boundary integral equation methods are used to solve problems in elastostatics. The main equations related to formulation of the boundary integral equation and the boundary element methods in 2D and 3D elastostatics are discussed in details. For their regularization, an approach based on the theory of distribution and the application of the Green theorem has been used. The expressions, which allow an easy calculation of the weakly singular, singular, and hypersingular integrals, have been constructed.

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7

Abdillah, Muhammad Taufik, Berlian Setiawaty, and Sugi Guritman. "The Solution of Generalization of the First and Second Kind of Abel’s Integral Equation." JTAM (Jurnal Teori dan Aplikasi Matematika) 7, no. 3 (July 17, 2023): 631. http://dx.doi.org/10.31764/jtam.v7i3.14193.

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Integral equations are equations in which the unknown function is found to be inside the integral sign. N. H. Abel used the integral equation to analyze the relationship between kinetic energy and potential energy in a falling object, expressed by two integral equations. This integral equation is called Abel's integral equation. Furthermore, these equations are developed to produce generalizations and further generalizations for each equation. This study aims to explain generalizations of the first and second kind of Abel’s integral equations, and to find solution for each equation. The method used to determine the solution of the equation is an analytical method, which includes Laplace transform, fractional calculus, and manipulation of equation. When the analytical approach cannot solve the equation, the solution will be determined by a numerical method, namely successive approximations. The results showed that the generalization of the first kind of Abel’s integral equation solution can be determined using the Laplace transform method, fractional calculus, and manipulation of equation. On the other hand, the generalization of the second kind of Abel’s integral equation solution is obtained from the Laplace transform method. Further generalization of the first kind of Abel’s integral equation solution can be obtained using manipulation of equation method. Further generalization of the second kind of Abel’s integral equation solution cannot be determined by analytical method, so a numerical method (successive approximations) is used.

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8

Chen, Jeng-Tzong, Chia-Chun Hsiao, and Shyue-Yuh Leu. "Null-Field Integral Equation Approach for Plate Problems With Circular Boundaries." Journal of Applied Mechanics 73, no. 4 (October 18, 2005): 679–93. http://dx.doi.org/10.1115/1.2165239.

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In this paper, a semi-analytical approach for circular plate problems with multiple circular holes is presented. Null-field integral equation is employed to solve the plate problems while the kernel functions in the null-field integral equation are expanded to degenerate kernels based on the separation of field and source points in the fundamental solution. The unknown boundary densities of the circular plates are expressed in terms of Fourier series. It is noted that all the improper integrals are transformed to series sum and are easily calculated when the degenerate kernels and Fourier series are used. By matching the boundary conditions at the collocation points, a linear algebraic system is obtained. After determining the unknown Fourier coefficients, the displacement, slope, normal moment, and effective shear force of the plate can be obtained by using the boundary integral equations. Finally, two numerical examples are proposed to demonstrate the validity of the present method and the results are compared with the available exact solution, the finite element solution using ABAQUS software and the data of Bird and Steele.

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9

Saadeh, Rania. "Applications of Double ARA Integral Transform." Computation 10, no. 12 (December 8, 2022): 216. http://dx.doi.org/10.3390/computation10120216.

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This paper describes our construction of a new double transform, which we call the double ARA transform (DARAT). Our novel double-integral transform can be used to solve partial differential equations and other problems. We discuss some fundamental characteristics of our approach, including existence, linearity, and several findings relating to partial derivatives and the double convolution theorem. DARAT can be used to precisely solve a variety of partial differential equations, including the heat equation, wave equation, telegraph equation, Klein–Gordon equation, and others, all of which are crucial for physical applications. Herein, we use DARAT to solve model integral equations to obtain exact solutions. We conclude that our novel method is easier to use than comparable transforms.

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10

Avdonin, S. A., B. P. Belinskiy, and John V. Matthews. "Inverse problem on the semi-axis: local approach." Tamkang Journal of Mathematics 42, no. 3 (August 24, 2011): 275–93. http://dx.doi.org/10.5556/j.tkjm.42.2011.916.

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We consider the problem of reconstruction of the potential for the wave equation on the semi-axis. We use the local versions of the Gelfand-Levitan and Krein equations, and the linear version of Simon's approach. For all methods, we reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand-side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples verify that the algorithms can be used to reconstruct an unknown potential accurately. The practicality of each approach is briefly discussed. Accurate data preparation is described and implemented.

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11

Brugnano, Luigi, Gianluca Frasca-Caccia, and Felice Iavernaro. "Line Integral Solution of Hamiltonian PDEs." Mathematics 7, no. 3 (March 18, 2019): 275. http://dx.doi.org/10.3390/math7030275.

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In this paper, we report on recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs) by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schrödinger equation, and the Korteweg–de Vries equation, to illustrate the main features of this novel approach.

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12

Alesemi, Meshari. "An Innovative Approach to Nonlinear Fractional Shock Wave Equations Using Two Numerical Methods." Mathematics 11, no. 5 (March 5, 2023): 1253. http://dx.doi.org/10.3390/math11051253.

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In this research, we propose a combined approach to solving nonlinear fractional shock wave equations using an Elzaki transform, the homotopy perturbation method, and the Adomian decomposition method. The nonlinear fractional shock wave equation is first transformed into an equivalent integral equation using the Elzaki transform. The homotopy perturbation method and Adomian decomposition method are then utilized to approximate the solution of the integral equation. To evaluate the effectiveness of the proposed method, we conduct several numerical experiments and compare the results with existing methods. The numerical results show that the combined method provides accurate and efficient solutions for nonlinear fractional shock wave equations. Overall, this research contributes to the development of a powerful tool for solving nonlinear fractional shock wave equations, which has potential applications in many fields of science and engineering. This study presents a solution approach for nonlinear fractional shock wave equations using a combination of an Elzaki transform, the homotopy perturbation method, and the Adomian decomposition method. The Elzaki transform is utilized to transform the nonlinear fractional shock wave equation into an equivalent integral equation. The homotopy perturbation method and Adomian decomposition method are then employed to approximate the solution of the integral equation. The effectiveness of the combined method is demonstrated through several numerical examples and compared with other existing methods. The results show that the proposed method provides accurate and efficient solutions for nonlinear fractional shock wave equations.

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13

Bayman, B. F., and F. Zardi. "Integral equation approach to relativistic Coulomb excitation." Physical Review C 59, no. 4 (April 1, 1999): 2189–208. http://dx.doi.org/10.1103/physrevc.59.2189.

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14

Davì, Giuseppe, and Alberto Milazzo. "Integral equation approach to composite laminate analysis." Journal of the Chinese Institute of Engineers 22, no. 6 (September 1999): 695–708. http://dx.doi.org/10.1080/02533839.1999.9670506.

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15

Douglas, Jack F. "Integral equation approach to condensed matter relaxation." Journal of Physics: Condensed Matter 11, no. 10A (January 1, 1999): A329—A340. http://dx.doi.org/10.1088/0953-8984/11/10a/030.

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16

Iemma, Umberto, and Lorenzo Burghignoli. "An integral equation approach to acoustic cloaking." Journal of Sound and Vibration 331, no. 21 (October 2012): 4629–43. http://dx.doi.org/10.1016/j.jsv.2012.04.032.

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17

Lamb, W. "A spectral approach to an integral equation." Glasgow Mathematical Journal 26, no. 1 (January 1985): 83–89. http://dx.doi.org/10.1017/s0017089500005802.

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In a recent paper [7], Rooney used a technique involving the Mellin transform to obtain solutions in certain spaces ℒμ, ρ of an integral equation which had been studied previously by Šub-Sizonenko [9]. The integral equation in question can be written aswhere I denotes the identity operator and G0.1/2 is given bywith the inversion formula obtained by Rooney taking the formRooney verified that (1.1) and (1.2) formed an inversion pair in ℒμ, ρ for 1 ≤ p < ∞ and μ > 0.

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18

Journal, Baghdad Science. "Solution of Variavle Delay integral eqiations using Variational approach." Baghdad Science Journal 3, no. 3 (September 3, 2006): 481–87. http://dx.doi.org/10.21123/bsj.3.3.481-487.

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The main objective of this research is to use the methods of calculus ???????? solving integral equations Altbataah When McCann slowdown is a function of time as the integral equation used in this research is a kind of Volterra

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19

Akishin, Pavel G., and Andrey A. Sapozhnikov. "The volume integral equation method in magnetostatic problem." Discrete and Continuous Models and Applied Computational Science 27, no. 1 (December 15, 2019): 60–69. http://dx.doi.org/10.22363/2658-4670-2019-27-1-60-69.

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This article addresses the issues of volume integral equation method application to magnetic system calculations. The main advantage of this approach is that in this case finding the solution of equations is reduced to the area filled with ferromagnetic. The difficulty of applying the method is connected with kernel singularity of integral equations. For this reason in collocation method only piecewise constant approximation of unknown variables is used within the limits of fragmentation elements inside the famous package GFUN3D. As an alternative approach the points of observation can be replaced by integration over fragmentation element, which allows to use approximation of unknown variables of a higher order.In the presented work the main aspects of applying this approach to magnetic systems modelling are discussed on the example of linear approximation of unknown variables: discretisation of initial equations, decomposition of the calculation area to elements, calculation of discretised system matrix elements, solving the resulting nonlinear equation system. In the framework of finite element method the calculation area is divided into a set of tetrahedrons. At the beginning the initial area is approximated by a combination of macro-blocks with a previously constructed two-dimensional mesh at their borders. After that for each macro-block separately the procedure of tetrahedron mesh construction is performed. While calculating matrix elements sixfold integrals over two tetrahedra are reduced to a combination of fourfold integrals over triangles, which are calculated using cubature formulas. Reduction of singular integrals to the combination of the regular integrals is proposed with the methods based on the concept of homogeneous functions. Simple iteration methods are used to solve non-linear discretized systems, allowing to avoid reversing large-scale matrixes. The results of the modelling are compared with the calculations obtained using other methods.

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20

Ghamkhar, Madiha, Laiba Wajid, Khurrem Shahzad, Rabia Safdar, Sharifah E. Alhazmi, Maha M. A. Lashin, Wasim Jamshed, Mohamed R. Eid, and Soliman Alkhatib. "Approximate solution of linear integral equations by Taylor ordering method: Applied mathematical approach." Open Physics 20, no. 1 (January 1, 2022): 850–58. http://dx.doi.org/10.1515/phys-2022-0182.

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Abstract Since obtaining an analytic solution to some mathematical and physical problems is often very difficult, academics in recent years have focused their efforts on treating these problems using numerical methods. In science and engineering, systems of integral differential equations and their solutions are extremely important. The Taylor collocation method is described as a matrix approach for solving numerically Linear Differential Equations (LDE) by using truncated Taylor series. Integral equations are used to solve problems such as radiative transmission and the oscillation of a string, membrane, or axle. Differential equations can be used to tackle oscillating difficulties. To discover approximate solutions for linear systems of integral differential equations with variable coefficients in terms of Taylor polynomials, the collocation approach, which is offered for differential and integral equation solutions, will be developed. A system of LDE will be translated into matrix equations, and a new matrix equation will be generated in terms of the Taylor coefficients matrix by employing Taylor collocation points. The needed system will be converted to a linear algebraic equation system. Finding the Taylor coefficients will lead to the Taylor series technique.

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21

Abas, Abas Wisam Mahdi. "The calculation of the solution of multidimensional integral equations with methods Monte Carlo and quasi-Monte Carlo." T-Comm 15, no. 10 (2021): 55–63. http://dx.doi.org/10.36724/2072-8735-2021-15-10-55-63.

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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.

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22

Girard, Ralph, and Alexander Langnau. "The light-cone approach for the calculation of hadronic states." Canadian Journal of Physics 67, no. 12 (December 1, 1989): 1162–67. http://dx.doi.org/10.1139/p89-195.

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We present an application of the recently proposed method of discretized light-cone quantization to quantum chromodynamics (QCD). We present the general method of light-cone quantization and point out its usefulness for the derivation of covariant integral equations for bound states. We then apply it to QCD and obtain the integral equation for the bound states in the valence sector of the mesons. The numerical solution of the integral equation shows a nice transition to the continuum. The conclusion summarizes the main results and discusses briefly what should be the next improvements of this method.

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23

Parnell, William J., and I. David Abrahams. "A new integral equation approach to elastodynamic homogenization." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2094 (February 26, 2008): 1461–82. http://dx.doi.org/10.1098/rspa.2007.0254.

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A new theory of elastodynamic homogenization is proposed, which exploits the integral equation form of Navier's equations and relationships between length scales within composite media. The scheme is introduced by focusing on its leading-order approximation for orthotropic, periodic fibre-reinforced media where fibres have arbitrary cross-sectional shape. The methodology is general but here it is shown for horizontally polarized shear (SH) wave propagation for ease of exposition. The resulting effective properties are shown to possess rich structure in that four terms account separately for the physical detail of the composite (associated with fibre cross-sectional shape, elastic properties, lattice geometry and volume fraction). In particular, the appropriate component of Eshelby's tensor arises naturally in order to deal with the shape of the fibre cross section. Results are plotted for circular fibres and compared with extant methods, including the method of asymptotic homogenization. The leading-order scheme is shown to be in excellent agreement even for relatively high volume fractions.

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24

Sun, Qiang, Evert Klaseboer, Boo-Cheong Khoo, and Derek Y. C. Chan. "Boundary regularized integral equation formulation of the Helmholtz equation in acoustics." Royal Society Open Science 2, no. 1 (January 2015): 140520. http://dx.doi.org/10.1098/rsos.140520.

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A boundary integral formulation for the solution of the Helmholtz equation is developed in which all traditional singular behaviour in the boundary integrals is removed analytically. The numerical precision of this approach is illustrated with calculation of the pressure field owing to radiating bodies in acoustic wave problems. This method facilitates the use of higher order surface elements to represent boundaries, resulting in a significant reduction in the problem size with improved precision. Problems with extreme geometric aspect ratios can also be handled without diminished precision. When combined with the CHIEF method, uniqueness of the solution of the exterior acoustic problem is assured without the need to solve hypersingular integrals.

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25

Sedeeg, Abdelilah Kamal, Zahra I. Mahamoud, and Rania Saadeh. "Using Double Integral Transform (Laplace-ARA Transform) in Solving Partial Differential Equations." Symmetry 14, no. 11 (November 15, 2022): 2418. http://dx.doi.org/10.3390/sym14112418.

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The main goal of this research is to present a new approach to double transforms called the double Laplace–ARA transform (DL-ARAT). This new double transform is a novel combination of Laplace and ARA transforms. We present the basic properties of the new approach including existence, linearity and some results related to partial derivatives and the double convolution theorem. To obtain exact solutions, the new double transform is applied to several partial differential equations such as the Klein–Gordon equation, heat equation, wave equation and telegraph equation; each of these equations has great utility in physical applications. In symmetry to other symmetric transforms, we conclude that our new approach is simpler and needs less calculations.

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26

Belov, A. A., and A. N. Petrov. "NUMERICAL ANALYSIS OF THE DYNAMICS OF THREE-DIMENSIONAL ANISOTROPIC BODIES BASED ON NON-CLASSICAL BOUNDARY INTEGRAL EQUATIONS." Problems of strenght and plasticity 83, no. 1 (2021): 76–86. http://dx.doi.org/10.32326/1814-9146-2021-83-1-76-86.

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The application of non-classical approach of the boundary integral equation method in combination with the integral Laplace transform in time to anisotropic elastic wave modeling is considered. In contrast to the classical approach of the boundary integral equation method which is successfully implemented for solving three-dimensional isotropic problems of the dynamic theory of elasticity, viscoelasticity and poroelasticity, the alternative nonclassical formulation of the boundary integral equations method is presented that employs regular Fredholm integral equations of the first kind (integral equations on a plane wave). The construction of such boundary integral equations is based on the structure of the dynamic fundamental solution. The approach employs the explicit boundary integral equations. The inverse Laplace transform is constructed numerically by the Durbin method. A numerical solution of the dynamic problem of anisotropic elasticity theory based on the boundary integral equations method in a nonclassical formulation is presented. The boundary element scheme of the boundary integral equations method is built on the basis of a regular integral equation of the first kind. The problem is solved in anisotropic formulation for the load acting along the normal in the form of the Heaviside function on the cube face weakened by a cubic cavity. The obtained boundary element solutions are compared with finite element solutions. Numerical results prove the efficiency of using boundary integral equations on a single plane wave in solving three-dimensional anisotropic dynamic problems of elasticity theory. The convergence of boundary element solutions is studied on three schemes of surface discretization. The achieved calculation accuracy is not inferior to the accuracy of boundary element schemes for classical boundary integral equations. Boundary element analysis of solutions for a cube with and without a cavity is carried out.

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27

Das, Ashok K., Sudhakar Panda, and J. R. L. Santos. "A path integral approach to the Langevin equation." International Journal of Modern Physics A 30, no. 07 (March 5, 2015): 1550028. http://dx.doi.org/10.1142/s0217751x15500281.

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We study the Langevin equation with both a white noise and a colored noise. We construct the Lagrangian as well as the Hamiltonian for the generalized Langevin equation which leads naturally to a path integral description from first principles. This derivation clarifies the meaning of the additional fields introduced by Martin, Siggia and Rose in their functional formalism. We show that the transition amplitude, in this case, is the generating functional for correlation functions. We work out explicitly the correlation functions for the Markovian process of the Brownian motion of a free particle as well as for that of the non-Markovian process of the Brownian motion of a harmonic oscillator (Uhlenbeck–Ornstein model). The path integral description also leads to a simple derivation of the Fokker–Planck equation for the generalized Langevin equation.

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28

Zhou, Yaoqi, and George Stell. "Fluids inside a pore—an integral-equation approach." Molecular Physics 66, no. 4 (March 1989): 767–89. http://dx.doi.org/10.1080/00268978900100511.

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Zhou, Yaoqi, and George Stell. "Fluids inside a pore—an integral-equation approach." Molecular Physics 66, no. 4 (March 1989): 791–96. http://dx.doi.org/10.1080/00268978900100521.

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Zhou, Yaoqi, and George Stell. "Fluids inside a pore—An integral equation approach." Molecular Physics 68, no. 6 (December 20, 1989): 1265–75. http://dx.doi.org/10.1080/00268978900102891.

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31

Curro, John G., and Kenneth S. Schweizer. "Theory of polymer melts: an integral equation approach." Macromolecules 20, no. 8 (August 1987): 1928–34. http://dx.doi.org/10.1021/ma00174a040.

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32

Chen, Z. Q., and F. J. Paoloni. "An integral equation approach to electrical conductance tomography." IEEE Transactions on Medical Imaging 11, no. 4 (1992): 570–76. http://dx.doi.org/10.1109/42.192693.

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33

Thye-Lai Tung and D. A. Antoniadis. "A boundary integral equation approach to oxidation modeling." IEEE Transactions on Electron Devices 32, no. 10 (October 1985): 1954–59. http://dx.doi.org/10.1109/t-ed.1985.22227.

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Thye-Lai Tung and D. A. Antoniadis. "A Boundary Integral Equation Approach to Oxidation Modeling." IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 4, no. 4 (October 1985): 398–403. http://dx.doi.org/10.1109/tcad.1985.1270137.

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35

Sloss, J. M., J. C. Bruch, S. Adali, and I. S. Sadek. "Piezoelectric patch control using an integral equation approach." Thin-Walled Structures 39, no. 1 (January 2001): 45–63. http://dx.doi.org/10.1016/s0263-8231(00)00053-7.

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36

Lo, S. H., C. Y. Dong, and Y. K. Cheung. "Integral equation approach for 3D multiple-crack problems." Engineering Fracture Mechanics 72, no. 12 (August 2005): 1830–40. http://dx.doi.org/10.1016/j.engfracmech.2004.11.009.

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37

Baer, Michael. "Integral equation approach to atom-diatom exchange processes." Physics Reports 178, no. 3 (June 1989): 99–143. http://dx.doi.org/10.1016/0370-1573(89)90137-3.

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38

Dobler, Wolfgang, and Karl-Heinz Rädler. "An integral equation approach to kinematic dynamo models." Geophysical & Astrophysical Fluid Dynamics 89, no. 1-2 (November 1998): 45–74. http://dx.doi.org/10.1080/03091929808213648.

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39

Ahmed, Salim. "Identification from step response - The integral equation approach." Canadian Journal of Chemical Engineering 94, no. 12 (October 10, 2016): 2243–56. http://dx.doi.org/10.1002/cjce.22645.

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40

Kaplan, Meydan, and Yaniv Brick. "A fast solver framework for acoustic hybrid integral equations." Journal of the Acoustical Society of America 152, no. 4 (October 2022): A119. http://dx.doi.org/10.1121/10.0015743.

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Reliable modeling of the scattering by acoustically large and geometrically complex objects can be achieved by means of subdomain-dependent problem formulation and a numerically rigorous solution. While the objects’ inhomogeneity has driven the development of differential equation formulations and solvers, integral equation formulations, where the object’s background is modeled via a Green’s function, are advantageous for unbounded domains. In the hybrid integral equations approach (Usner et al., 2006), the interaction of separate subdomains with external fields is described by pertinent integral equations. Their Galerkin discretization leads to a dense blocked stiffness matrix. The development of compressed representations of the matrix, which are necessary for the treatment of large systems, becomes non-trivial due to the multitude of integral equation kernels and the different geometrical and physical characteristic of the subdomains. As part of the development of a fast hybrid integral equation solver framework, we consider the case of objects composed of large inhomogeneous volumes, modeled as incompressible fluids, and of simplified solids, modeled via surface integral equations. A hybrid integral equation formulation is derived and solved numerically. The iterative solution is accelerated by employing the butterfly-compressed hierarchical representation of the stiffness matrix, recently used for acoustic volume integral equations (Kaplan, 2022).

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Yépez-Martínez, Huitzilin, Ivan O. Sosa, and Juan M. Reyes. "Feng’s First Integral Method Applied to the ZKBBM and the Generalized Fisher Space-Time Fractional Equations." Journal of Applied Mathematics 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/191545.

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The fractional derivatives in the sense of the modified Riemann-Liouville derivative and Feng’s first integral method are employed to obtain the exact solutions of the nonlinear space-time fractional ZKBBM equation and the nonlinear space-time fractional generalized Fisher equation. The power of this manageable method is presented by applying it to the above equations. Our approach provides first integrals in polynomial form with high accuracy. Exact analytical solutions are obtained through establishing first integrals. The present method is efficient and reliable, and it can be used as an alternative to establish new solutions of different types of fractional differential equations applied in mathematical physics.

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Kirsch, Andreas. "An integral equation approach and the interior transmission problem for Maxwell's equations." Inverse Problems & Imaging 1, no. 1 (2007): 159–79. http://dx.doi.org/10.3934/ipi.2007.1.159.

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Klaseboer, Evert, Qiang Sun, and Derek Y. C. Chan. "Non-singular boundary integral methods for fluid mechanics applications." Journal of Fluid Mechanics 696 (March 7, 2012): 468–78. http://dx.doi.org/10.1017/jfm.2012.71.

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AbstractA formulation of the boundary integral method for solving partial differential equations has been developed whereby the usual weakly singular integral and the Cauchy principal value integral can be removed analytically. The broad applicability of the approach is illustrated with a number of problems of practical interest to fluid and continuum mechanics including the solution of the Laplace equation for potential flow, the Helmholtz equation as well as the equations for Stokes flow and linear elasticity.

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Zakerdoost, Hassan, Hassan Ghassemi, and Mehdi Iranmanesh. "Solution of Boundary Value Problems Using Dual Reciprocity Boundary Element Method." Advances in Applied Mathematics and Mechanics 9, no. 3 (January 17, 2017): 680–97. http://dx.doi.org/10.4208/aamm.2014.m783.

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AbstractIn this work we utilize the boundary integral equation and the Dual Reciprocity Boundary Element Method (DRBEM) for the solution of the steady state convection-diffusion-reaction equations with variable convective coefficients in two-dimension. The DRBEM is a numerical method to transform the domain integrals into the boundary only integrals by using the fundamental solution of Helmholtz equation. Some examples are calculated to confirm the accuracy of the approach. The results obtained by the analytic solutions are in good agreement with ones provided by the DRBEM technique.

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CHEN, J. T., C. T. CHEN, and I. L. CHEN. "NULL-FIELD INTEGRAL EQUATION APPROACH FOR EIGENPROBLEMS WITH CIRCULAR BOUNDARIES." Journal of Computational Acoustics 15, no. 04 (December 2007): 401–28. http://dx.doi.org/10.1142/s0218396x07003391.

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In this paper, the eigenproblems with circular boundaries of multiply-connected domain are studied by using the null-field integral equations in conjunction with degenerate kernels and Fourier series to avoid calculating the Cauchy and Hadamard principal values. An adaptive observer system of polar coordinate is considered to fully employ the property of degenerate kernels. For the hypersingular equation, vector decomposition for the radial and tangential gradients is carefully considered in the polar coordinate system. Direct-searching scheme is employed to detect the eigenvalues by using the singular value decomposition (SVD) technique. Both the singular and hypersingular equations result in spurious eigenvalues which are the associated interior Dirichlet and Neumann problems of interior domain of inner circles, respectively. It is analytically verified that the spurious eigenvalue depends on the radius of any inner circle and numerical experiments support this point. Several examples are demonstrated to see the validity of the present formulation. More number of degrees of freedom of BEM is required to obtain the same accuracy of the present approach.

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Jung, Soon-Mo. "A Fixed Point Approach to the Stability of an Integral Equation Related to the Wave Equation." Abstract and Applied Analysis 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/612576.

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Zhang, Jian, Jinjiao Hou, Jing Niu, Ruifeng Xie, and Xuefei Dai. "A high order approach for nonlinear Volterra-Hammerstein integral equations." AIMS Mathematics 7, no. 1 (2021): 1460–69. http://dx.doi.org/10.3934/math.2022086.

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<abstract><p>Here a scheme for solving the nonlinear integral equation of Volterra-Hammerstein type is given. We combine the related theories of homotopy perturbation method (HPM) with the simplified reproducing kernel method (SRKM). The nonlinear system can be transformed into linear equations by utilizing HPM. Based on the SRKM, we can solve these linear equations. Furthermore, we discuss convergence and error analysis of the HPM-SRKM. Finally, the feasibility of this method is verified by numerical examples.</p></abstract>

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Chakrabarti, A., Sudeshna Banerjea, B. N. Mandal, and T. Sahoo. "A unified approach to problems of scattering of surface water waves by vertical barriers." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 39, no. 1 (July 1997): 93–103. http://dx.doi.org/10.1017/s0334270000009231.

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AbstractA unified analysis involving the solution of multiple integral equations via a simple singular integral equation with a Cauchy type kernel is presented to handle problems of surface water wave scattering by vertical barriers. Some well known results are produced in a simple and systematic manner.

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Malyshev, Igor. "Surface integrals approach to solution of some free boundary problems - II." Journal of Applied Mathematics and Simulation 2, no. 2 (January 1, 1989): 91–100. http://dx.doi.org/10.1155/s1048953389000079.

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This paper is a continuation of the publication [1] where integral equation techniques were applied to the solution of a generalized Stefan problem. The regularization of the corresponding system of nonlinear integral Volterra equations offered here is quite different from that in [1], hence - several new algorithms and numerical experiments. For consistency and easy reference we start this paper with sec.6.

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Mallardo, Vincenzo, Eugenio Ruocco, and Gernot Beer. "A NURBS-BEM Application in Continuum Damage Mechanics." Key Engineering Materials 774 (August 2018): 253–58. http://dx.doi.org/10.4028/www.scientific.net/kem.774.253.

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In this paper the integral equation approach is used to describe the propagation of continuumdamage in three dimensional solids. The governing equation is of integral type and contains bothboundary and domain integrals. Such integrals are computed with the aid of the NURBS functions.The subvolume involved by the damage is modelled by a special mapping procedure that avoids theuse of the internal cells. The implementation is verified on a test case for which an analytical solutionis available.

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