This book examines the basic mathematical properties of solutions to boundary integral equations and details the variational methods for the boundary integral equations arising in elasticity, fluid ...mechanics and acoustic scattering theory.
A new numerical method based on Haar wavelet is proposed for two-dimensional nonlinear Fredholm, Volterra and Volterra–Fredholm integral equations of first and second kind. The proposed method is an ...extension of the Haar wavelet method Aziz and Siraj-ul-Islam (2013), Siraj-ul-Islam et al. (2013) and Siraj-ul-Islam et al. (2014) from one-dimensional nonlinear integral equations (Fredholm and Volterra) to two-dimensional nonlinear integral equations (Fredholm, Volterra and Volterra–Fredholm). The main characteristic of the method is that, unlike several other methods, it does not involve numerical integration which results in an improved accuracy of the method. In order to show the effectiveness of the method, it is applied to several benchmark problems. The numerical results are compared with other methods existing in the recent literature.
In this manuscript, the aim is to prove a multiple fixed point (FP) result for partially ordered s-distance spaces under θ,ϕ,ψ-type weak contractive condition. The result will generalize some ...well-known results in literature such as coupled FP (Guo and Lakshmikantham, 1987), triple fixed point (Berinde and Borcut, 2011), and quadruple FP results (Karapinar, 2011). Moreover, to validate the result, an application for the existence of solution of a system of integral equations is also provided.
In this paper, orthonormal Bernoulli collocation method has been developed to obtain the approximate solution of linear singular stochastic Itô‐Volterra integral equations. By applying this method, ...linear stochastic integral equation converts to linear system of algebraic equations. This system is achieved by approximating functions that appear in the stochastic integral equations by using orthonormal Bernoulli polynomials (OBPs) and then substituting these approximations into consideration equation. This linear system of algebraic equations can be solved via an appropriate numerical method and approximate solution of integral equation is obtained. A main advantage of this technique is that the condition number of the coefficient matrix of the system is small, which verify that THE proposed method is stable. Also, convergence and error analysis of the present method are discussed. Finally, two examples are given to show the pertinent properties, applicability, and accuracy of the present method.
A butterfly-accelerated volume integral equation (VIE) solver is proposed for fast and accurate electromagnetic (EM) analysis of scattering from heterogeneous objects. The proposed solver leverages ...the hierarchical off-diagonal butterfly (HOD-BF) scheme to construct the system matrix and obtain its approximate inverse, used as a preconditioner. Complexity analysis and numerical experiments validate the <inline-formula> <tex-math notation="LaTeX">O(N\log ^{2}N) </tex-math></inline-formula> construction cost of the HOD-BF-compressed system matrix and <inline-formula> <tex-math notation="LaTeX">O(N^{1.5}\log N) </tex-math></inline-formula> inversion cost for the preconditioner, where <inline-formula> <tex-math notation="LaTeX">N </tex-math></inline-formula> is the number of unknowns in the high-frequency EM scattering problem. For many practical scenarios, the proposed VIE solver requires less memory and computational time to construct the system matrix and obtain its approximate inverse compared to a <inline-formula> <tex-math notation="LaTeX">\mathcal {H} </tex-math></inline-formula> matrix-accelerated VIE solver. The accuracy and efficiency of the proposed solver have been demonstrated via its application to the EM analysis of large-scale canonical and real-world structures comprising of broad permittivity values and involving millions of unknowns.
Purpose
This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and ...the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm.
Design/methodology/approach
The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations.
Findings
Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software.
Research limitations/implications
The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement.
Practical implications
There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively.
Social implications
This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution.
Originality/value
To the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.
In this article, we use a fuzzy number in its parametric form to solve a fuzzy nonlinear integral equation of the second kind in the crisp case. The main theme of this article is to find a ...semi-analytical solution of fuzzy nonlinear integral equations. A hybrid method of Laplace transform coupled with Adomian decomposition method is used to find the solution of the fuzzy nonlinear integral equations including fuzzy nonlinear Fredholm integral equation, fuzzy nonlinear Volterra integral equation, and fuzzy nonlinear singular integral equation of Abel type kernel. We also provide some suitable examples to better understand the proposed method.
This paper presents a computational method based on the Chebyshev wavelets for solving stochastic Itô–Volterra integral equations. First, a stochastic operational matrix for the Chebyshev wavelets is ...presented and a general procedure for forming this matrix is given. Then, the Chebyshev wavelets basis along with this stochastic operational matrix are applied for solving stochastic Itô–Volterra integral equations. Convergence and error analysis of the Chebyshev wavelets basis are investigated. To reveal the accuracy and efficiency of the proposed method some numerical examples are included.
In this paper we prove some results concerning the existence of solutions for a large class of nonlinear Volterra singular integral equations in the space
C
0
,
1
consisting of real functions ...defined and continuous on the interval
0
,
1
. The main tool used in the proof is the concept of a measure of noncompactness. We also present some examples of nonlinear singular integral equations of Volterra type to show the efficiency of our results. Moreover, we compare our theory with the approach depending on the use of the theory of Volterra–Stieltjes integral equations. We also show that the results of the paper are applicable in the study of the so-called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems.
In this paper, a new computational method based on the generalized hat basis functions is proposed for solving stochastic Itô–Volterra integral equations. In this way, a new stochastic operational ...matrix for generalized hat functions on the finite interval 0,T is obtained. By using these basis functions and their stochastic operational matrix, such problems can be transformed into linear lower triangular systems of algebraic equations which can be directly solved by forward substitution. Also, the rate of convergence of the proposed method is considered and it has been shown that it is O(1n2). Further, in order to show the accuracy and reliability of the proposed method, the new approach is compared with the block pulse functions method by some examples. The obtained results reveal that the proposed method is more accurate and efficient in comparison with the block pule functions method.
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