Explore the vast range of titles available.

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.

The authors consider the nonlinear equation -\\frac 1m=z+Sm with a parameter z in the complex upper half plane \\mathbb H , where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in \\mathbb H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on \\mathbb R. In a previous paper the authors qualitatively identified the possible singular behaviors of v: under suitable conditions on S we showed that in the density of v only algebraic singularities of degree two or three may occur. In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any z\\in \\mathbb H, including the vicinity of the singularities.

This article is concerned with the question of uniqueness of self-similar profiles for Smoluchowski’s coagulation equation which exhibit algebraic decay (fat tails) at infinity. More precisely, we consider a rate kernel Establishing uniqueness of self-similar profiles for Smoluchowski’s coagulation equation is generally considered to be a difficult problem which is still essentially open. Concerning fat-tailed self-similar profiles this article actually gives the first uniqueness statement for a non-solvable kernel.

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.

In this paper, we prove both necessary and sufficient maximum principles for infinite horizon discounted control problems of stochastic Volterra integral equations with finite delay and a convex control domain. The corresponding adjoint equation is a novel class of infinite horizon anticipated backward stochastic Volterra integral equations. Our results can be applied to discounted control problems of stochastic delay differential equations and fractional stochastic delay differential equations. As an example, we consider a stochastic linear-quadratic regulator problem for a delayed fractional system. Based on the maximum principle, we prove the existence and uniqueness of the optimal control for this concrete example and obtain a new type of explicit Gaussian state-feedback representation formula for the optimal control.

In this manuscript, some fixed point results for generalized contractive type mappings under mild conditions in the setting of double controlled metric spaces (in short, ηℷν-metric spaces) are obtained. Moreover, some related consequences dealing with a common fixed point concept and nontrivial examples to support our results are presented. Ultimately, we use the theoretical results to discuss the existence and uniqueness of solutions of 2D Volterra integral equations, Riemann–Liouville integrals and Atangana–Baleanu integral operators are given.

This paper investigates functional integral equations with multivalued terms that appear symmetrically on both sides of the equation. We impose structural conditions on the coefficients that, while insufficient to ensure uniqueness, are adequate to guarantee the existence of at least one solution. We assumed mere continuity of the coefficients to establish a Peano-type existence theorem. The existence result was derived through the application of Schauder’s fixed-point theorem. We further highlighted that this finding for symmetric multivalued functional integral equations significantly informs the analysis of delayed multivalued functional equations, which is commonly utilized in modeling processes where both the present and past states of the system play a crucial role. To illustrate the applicability of the proposed frameworks, representative examples were formulated and numerically solved for each type of equation, highlighting potential use cases in medicine and logistics.

The present paper aims to define three new notions: Θ e -contraction, a Hardy–Rogers-type Θ -contraction, and an interpolative Θ -contraction in the framework of extended b-metric space. Further, some fixed point results via these new notions and the study endeavors toward a feasible solution would be suggested for nonlinear Volterra–Fredholm integral equations of certain types, as well as a solution to a nonlinear fractional differential equation of the Caputo type by using the obtained results. It also considers a numerical example to indicate the effectiveness of this new technique.

The goal of this article is to offer a series of results related to the existence and properties of wavefront solutions for doubly nonlinear diffusion–reaction equations involving the p -Laplacian operator in terms of the constitutive functions of the problem. These results are derived from the analysis of singular Volterra integral equations that appear in the study of monotone travelling-wave solutions for such equations. Our results extend the ones due to B. Gilding and R. Kersner for the case p = 2 to p > 1 . The fact that p ≠ 2 modifies the nature of the singularity in the integral equation, and introduces the need to develop some new tools and ideas for the analysis.

In this letter, a Quasi‐Helmholtz Projector method including multibranch Rao‐Wilton‐Glisson (MB‐RWG) basis function to calculate the scattering problem of multiscale targets at low frequencies is proposed. The loop basis function and the star basis function including MB‐RWG basis function are constructed, and the loop basis function and the star basis function including MB‐RWG basis function are used to construct the Quasi‐Helmholtz Projector. The proposed method is coined MB‐Quasi‐Helmholtz Projector (MB‐QHP) method. The MB‐QHP method can effectively solve the low frequency breakdown (LFB) problem in electric field integral equation (EFIE). Compared with the MB‐loop‐star method, the condition number is lower, the convergence is faster, and the search for global loop is avoided. Meanwhile, due to the existence of the MB‐RWG basis function, the proposed method can divide the calculation region into several regions with different mesh sizes. Numerical example shows the advantages of the proposed method. In this letter, a Quasi‐Helmholtz Projector method is proposed including multibranch Rao‐Wilton‐Glisson (MB‐RWG) basis function to calculate the scattering problem of multiscale targets at low frequencies.