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Green function of a 2n-th order differential equation with normal coefficients on the half-axis is studied. We first consider the Green function of our equation with “frozen” coefficients. Using Levi’s method, we obtain a Fredholm-type integral equation for the Green function of our problem, whose kernel is a Green function of a problem with constant coefficients. We prove an existence and uniqueness theorem for this integral equation in some Banach spaces of operator-valued functions. The main result of this paper is a theorem stating that the solution of the obtained integral equation is a Green function of our problem. [ABSTRACT FROM AUTHOR]

Как хорошо известно, наличие характеристик является очень существенным при исследовании задачи Коши для дифференциальных уравнений с частными производными независимо от его порядка. В случае, если дифференциальное уравнение с частными производными является нагруженным, то для однозначной разрешимости задачи Коши возникают дополнительные условия разрешимости, зависящие от вида следа нагрузки. Эти условия возникают даже для простейших линейных нагруженных дифференциальных уравнений с частными производными, начиная с первого порядка и выше. Основная цель данной работы – проиллюстрировать возникающие эффекты на примере исследования задачи Коши для линейного нагруженного уравнения в частных производных первого порядка. Так как корректность поставленной задачи Коши эквивалентным образом редуцируется к интегральному уравнению второго рода, то основной метод, применяемый для доказательства его разрешимости – метод последовательных подстановок. Основной вывод заключается в том, что разрешимость задачи Коши для нагруженного уравнения в частных производных существенным образом зависит от выбора следа нагрузки. В случае, когда разрешимость задачи Коши доказана, оказывается, что область влияния данных Коши не ограничивается только характеристиками, а появляются новые не характеристические линии, за которые данные Коши однозначно продолжаться не могут.

The existence and uniqueness theorems of the solution to the boundary-value problem for a parabolic-hyperbolic fractional-order equation with the gluing condition are proved. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the notion of neutrosophic pentagonal metric space. We prove several interesting results for some classes contraction mappings and prove some fixed point theorems in neutrosophic pentagonal metric space. Finally, we prove the uniqueness and existence of the integral equation and fractional differential equation to support our main result. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce a new integral transform named the W transform. We have been obtained some important results about the W transform. At the same time, the relation between the W transform and other transforms has been established. In order to prove the efficiency of this transform, we have solved the differential equations and integral equations. [ABSTRACT FROM AUTHOR]

In this article, we prove some novel fixed-point results for a pair of multivalued dominated mappings obeying a new generalized Nashine–Wardowski–Feng–Liu-type contraction for orbitally lower semi-continuous functions in a complete orbital b-metric space. Furthermore, some new fixed-point theorems for dominated multivalued mappings are established in the scenario of ordered complete orbital b-metric spaces. Some examples are offered to demonstrate the validity of our new results' premise. To demonstrate the applicability of our findings, applications for a system of nonlinear Volterra-type integral equations and fractional differential equations are shown. These results extend the theoretical results of Nashine et al. (Nonlinear Anal., Model. Control 26(3):522–533, 2021). [ABSTRACT FROM AUTHOR]

In this article, the recurrence relations and shift operators for multivariate Hermite polynomials are derived using the factorization approach. Families of differential equations, including differential, integro–differential, and partial differential equations, are obtained using these operators. The Volterra integral for these polynomials is also discovered. [ABSTRACT FROM AUTHOR]

Sonine kernel is characterized by the Sonine condition (denoted by SC) and is an important class of kernels in nonlocal differential equations and integral equations. This work proposes a SC with a more general form (denoted by gSC), which is more convenient than SC to accommodate complex kernels and equations. A typical kernel is given, and the first-kind Volterra integral equation under gSC is accordingly transformed and then analyzed. Based on these results, it is finally proved that the gSC is indeed equivalent to the original SC, which indicates that the Sonine kernel may be essentially characterized by the behavior of its convolution with the associated kernel at the starting point. [ABSTRACT FROM AUTHOR]

In this paper, parallel to the ideas based on Ahmadullah et al. [4, 5, 6], and Eke et al. [18], we prove the existence and uniqueness of the common fixed point for a pair of self-mappings employing (E. A.)-property in metric-like spaces for implicit contractive mappings related to binary relation. Henceforth, results obtained will be verified with the help of illustrative examples. As an application of the results, we solve two boundary value problems of the second-order differential equation. [ABSTRACT FROM AUTHOR]

Due to benefit of wavelets through numerical and other estimation methods and edge through Fourier analysis, the wavelet hypothesis has expanded broad significance at the time of previous years basically because of their application in comparing areas of science and masterminding, for instance, viscoelasticity, scattering of a natural people, signal taking care of, electromagnetism, fluid mechanics, electrochemistry, and some more. Wavelet has been fundamentally a wave design whose graph oscillates just through a short separation and dumps extremely quick. It tends to be utilized as equipment for taking care of such mathematical problems as differential conditions and integral issues. We have been utilizing wavelet techniques for fathoming the request differential condition; likewise, consider their accuracy and efficiency. [ABSTRACT FROM AUTHOR]

In this article, the recurrence relations and shift operators for multivariate Hermite polynomials are derived using the factorization approach. Families of differential equations, including differential, integro–differential, and partial differential equations, are obtained using these operators. The Volterra integral for these polynomials is also discovered.

The purpose of this manuscript is to demonstrate the existence and uniqueness of triple fixed-point results for Geraghty-type contractions in ordinary metric spaces with binary relations. Moreover, the well-posedness of the tripled fixed point problem is investigated. Consequently, α -dominated mapping on such space is discussed. Ultimately, to promote our paper, some illustrative examples are derived, and the existence of the solution to a system of differential equations is obtained as an application. [ABSTRACT FROM AUTHOR]

In the electrodynamic calculation of microwave (EHF) devices using methods that lead to algorithms in an open form, strict integral relations (representations) are very useful: Lorentz lemma, reciprocity theorem, orthogonality condition for eigenwaves, etc. of the results obtained, their convergence improves, and in some cases the calculation of characteristics that cannot be calculated without the indicated representations. Integral representations are a record of the equations of electrodynamics (in any unified form) and their solutions in one or another generalized form, linking in general the electromagnetic fields in electrodynamic structures described by boundary value problems. Integral views are used to control the results obtained; insome cases, they allow obtaining analytical solutions; lead to self-consistent problems that take into account the reverse effect of the radiation field on the primary sources; allow obtaining a priori information about the spectrum of possible solutions; solve associated problems as specific problems of arousal. Consideration of the phenomenon of complex resonance in this work shows that integral representations make it possible to establish a connection between non-self-adjointness and self-consistency of boundary value problems.

[Display omitted] • A new general integral transform which is covered all class of integral transform in the class of Laplace transform. • We investigated the application of this new transform for solving ODE with constant and variable coefficient. • This new transform can handle easily for fractional order integral equations and fractional order differential equations. • We have discussed the advantage and disadvantage of other integral transformed which is defined during last 2 decades. • We proved the related theorems for this new transform. Integral transforms are important to solve real problems. Appropriate choice of integral transforms helps to convert differential equations as well as integral equations into terms of an algebraic equation that can be solved easily. During last two decades many integral transforms in the class of Laplace transform are introduced such as Sumudu, Elzaki, Natural, Aboodh, Pourreza, Mohand, G_transform, Sawi and Kamal transforms. In this paper, we introduce a general integral transform in the class of Laplace transform. We study the properties of this transform. Then we compare it with few exiting integral transforms in the Laplace family such as Laplace, Sumudu, Elzaki and G\_transforms, Pourreza, Aboodh and etc. A new integral transform is introduced. Then some properties of this integral transform are discussed. This integral transform is used to solve this new transform is used for solving higher order initial value problems, integral equations and fractional order integral equation. It is proved that those new transforms in the class of Laplace transform which are introduced during last few decades are a special case of this general transform. It is shown that there is no advantage between theses transforms unless for special problems. It has shown that this new integral transform covers those exiting transforms such as Laplace, Elzaki and Sumudu transforms for different value of p (s) and q (s). We used this new transform for solving ODE, integral equations and fractional integral equations. Also, we can introduce new integral transforms by using this new general integral transform. [ABSTRACT FROM AUTHOR]

In this research, we study a weakly singular Volterra integro differential equation with Caputo‐type fractional derivative. First, we derive a sufficient condition for the existence and uniqueness of the solution of this problem based on the maximum norm. It is observed that the condition depends on the domain of definition of the problem. Thereafter, we show that this condition will be independent of the domain of definition based on an equivalent weighted maximum norm. In addition, we have also provided a procedure to extend the existence and uniqueness of the solution in its domain of definition by partitioning it. We also derive a sufficient condition under which the model problem will provide an analytic solution. Next, we introduce an operator‐based parameterized method to generate an approximate solution of this problem. Convergence analysis of this approach is established here. Next, we have optimized this solution based on least square method. For this, residual minimization is used to obtain the optimal values of the auxiliary parameter. In addition, we have also provided an error bound based on this technique. Several numerical examples are produced to clarify the effective behavior of the convergence of the present method. Comparison of the standard method and optimized method based on residual minimization signify the better accuracy of modified one. In addition, we also consider an equivalent form of weakly singular integro differential equation of Heat transfer problem to show the high effectiveness of the present approach. [ABSTRACT FROM AUTHOR]

The goal of this work is to look at generalised contractions and show that they may be used to derive fixed point theorems in generalised metric spaces. Furthermore, we'll also use fixed point theorems to show that the outcome of the ordinary differential equation exists and is exclusive (ODE). Fixed point theorem applies to modified Lipschitz condition. First, we have proved that generalised condition of Boyd Wong theorem and using this concept we have modified Lipschitz condition by adding constant which is suitable for solving differential equation. [ABSTRACT FROM AUTHOR]

The paper deals with the existence of three different weak solutions of p -Laplacian fractional for an overdetermined nonlinear fractional partial Fredholm–Volterra integro-differential system by using variational methods combined with a critical point theorem due to Bonanno and Marano. [ABSTRACT FROM AUTHOR]

Introduction/purpose: The aim of the paper is to establish some coupled fixed point results in C*-algebra-valued b-metric spaces. Moreover, the obtained results are used to define the sufficient conditions for the existence of the solutions of some classes of integral equations. Methods: The method of coupled fixed points gives the sufficient conditions for the existence of the solution of some classes of integral equations. Results: New results were obtained on coupled fixed points in C * -algebra-valued b-metric space. Conclusion: The obtained results represent a contribution in the fixed point theory and open new possibilities of application in the theory of differential and integral equations. [ABSTRACT FROM AUTHOR]

Integral equations of probability generating functions are generally recognized as very strong analytical tools of probability distributions theory. The present paper concentrates on the formulation, solution, theoretical and practical applicability of an integral equation for probability generating functions. [ABSTRACT FROM AUTHOR]

In this research work, we employ the critical point theory based on the variational structure to prove the existence of at least three distinct weak solutions for an overdetermined system of nonlinear fractional partial Fredholm-Volterra integro-differential equations. [ABSTRACT FROM AUTHOR]

Abstract In the first part of this work we show the convergence with respect to an asymptotic parameter ε of a delayed heat equation. It represents a mathematical extension of works considered previously by the authors [14–16]. Namely, this is the first result involving delay operators approximating protein linkages coupled with a spatial elliptic second order operator. For the sake of simplicity we choose the Laplace operator, although more general results could be derived. The main arguments are (i) new energy estimates and (ii) a stability result extended from the previous work to this more involved context. They allow to prove convergence of the delay operator to a friction term together with the Laplace operator in the same asymptotic regime considered without the space dependence in [14]. In a second part we extend fixed-point results for the fully non-linear model introduced in [16] and prove global existence in time. This shows that the blow-up scenario observed previously does not occur. Since the latter result was interpreted as a rupture of adhesion forces, we discuss the possibility of bond breaking both from the analytic and numerical point of view. [ABSTRACT FROM AUTHOR]

A Robin problem is a mixed problem with a linear combination of Dirichlet and Neumann D-N conditions. The aim of this paper are presents a new boundary integral equation BIE method for the solution of unbounded Robin boundary value problem BVP in the simply connected domain. The method show how to reformulate the Robin boundary value problem BVP as Riemann-Hilbert problem RHP which lead to the system of integral equation and the related differential equations are also created that give rise to unique solutions. Numerical results on several tests regions by the Nyström method NM with the trapezoidal rule TR are presented to clarify the solution technique for the Robin problem when the boundaries are sufficiently smooth. [ABSTRACT FROM AUTHOR]

In this paper we propose the study of a scalar integral equation of the type y(t) = g(y) + ∫∞ι (s - t)a(s)f(y(s)) ds; t ≥ 0; and give conditions on g, a and f that ensure the existence of solutions on [0;∞) which are asymptotically equal to g(y) at ∞. As a consequence, we obtain results on the existence of solutions for a problem of the type y"(t) = a(t)f(y(t)); y(∞) = g(y); where y(∞) = ... y(t). This problem could be thought as a sort of nonlocal problem at ∞, and our conditions on f include the case of a linear equation. [ABSTRACT FROM AUTHOR]

In this paper, some nonlinear Gronwall--Bellman type inequalities are established. Then, the obtained results are applied to study the Hyers--Ulam stability of a fractional differential equation and the boundedness of solutions to an integral equation, respectively. [ABSTRACT FROM AUTHOR]

In this paper we extend Darbo's fixed point theorem in Banach spaces and obtain a tripled fixed point theorem. The technique of measure of noncompactness is the main tool in carrying out our proof. Finally, as an application of our results, we analyze the existence of solutions for a system of neutral differential equations. [ABSTRACT FROM AUTHOR]

The paper deals with a boundary value problem for the Kirchhoff type static beam nonlinear integro-differential equation. The problem is reduced by Green function to an integral equation which is solved using the Picard iteration method. The convergence of the iteration process is established and numerical realization is obtained. [ABSTRACT FROM AUTHOR]

In this paper, by using the Guo-Krasnoselskii theorem, we investigate the existence and nonexistence of positive solutions of a system of integral equation with parameters which can be seen as an effective generalization of various types of systems of boundary value problems for differential equation on continuous interval and time scales or fractional differential equations. We give a general approach of positive solutions to cover various systems of boundary value problems in a unified way, which avoids treating these problems on a case-by-case basis. Under some growth conditions imposed on the nonlinear term, we obtain explicit ranges of values of parameters with which the problem has a positive solution and has no positive solution, respectively. By giving some examples, we will show how our results may be applied to consider existence of positive solutions to a variety of system of boundary value problems of differential equations, differential equations on time scales or fractional differential equations. [ABSTRACT FROM AUTHOR]

We use a modification of Krasnoselskii's fixed point theorem introduced by Burton to show the periodicity and non-negativity of solutions for the nonlinear neutral differential equation with variable delay [ABSTRACT FROM AUTHOR]

This paper presents a boundary integral equation method for finding the solution of Robin problems in bounded and unbounded multiply connected regions. The Robin problems are formulated as Riemann-Hilbert problems which lead to systems of integral equations and the related differential equations are also constructed that give rise to unique solutions, which are shown. Numerical results on several test regions are presented to illustrate that the approximate solution when using this method for the Robin problems when the boundaries are sufficiently smooth are accurate. [ABSTRACT FROM AUTHOR]

In this article we consider the following integral equation involving Bessel potentials on a half space ℝ+n: ... where α > 0, β > 1, ... is the reflection of x about xn = 0, and ga(x) denotes the Bessel kernel. We first enhance the regularity of positive solutions for the integral equation by regularity-lifting-method, which has been extensively used by many authors. Then, employing the method of moving planes in integral forms, we demonstrate that there is no positive solution for the integral equation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of multivalued interpolative Kannan-type contractions. We also introduce a more general version of this notion by relaxing the degrees of freedom of the powers arising in the contractive condition. Gaba et al. (2021) recently pointed out a significant error in the paper of Gaba and Karapinar (2019), showing that a particular type of generalized interpolative Kannan-type contraction does not posses a fixed point in general in a complete metric space. Thus, the study of generalized Kannan-type mappings remains an interesting and mathematically challenging area of research. The main aim of this article is to address such existing results for multivalued mappings. We also investigate common fixed points for this type of contractions. Our results extend and unify some existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we use a modification of Krasnoselskii's fixed point theorem introduced by Burton (2002) (see [6] Theorem 3) to obtain stability results of the zero solution of the totally nonlinear neutral differential equations with functional delay x′(t)=-a(t)h(x(t-τ(t)))+d/dtQ(t,x(t-τ(t)))+G(t,x(t-τ(t))). The stability of the zero solution of this equation provided that h(0)=Q(t,0)=G(t,0,0)=0. The Caratheodory condition is used for the functions Q and G. [ABSTRACT FROM AUTHOR]

In this study, properties of spectral characteristic are investigated for singular Sturm-Liouville operators in the case where an eigen parameter not only appears in the differential equation but is also linearly contained in the jump conditions. Also Weyl function for considering operator has been defined and the theorems which related to uniqueness of solution of inverse problem according to Weyl function and two spectra have been proved. [ABSTRACT FROM AUTHOR]

We obtain the optimal integrability for positive solutions of the Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality in ℝ: C. Jin and C. Li [Calc. Var. Partial Differential Equations, 2006, 26: 447-457] developed some very interesting method for regularity lifting and obtained the optimal integrability for p, q > 1. Here, based on some new observations, we overcome the difficulty there, and derive the optimal integrability for the case of p, q ⩾ 1 and pq ≠ 1. This integrability plays a key role in estimating the asymptotic behavior of positive solutions when | x| → 0 and when | x| → ∞. [ABSTRACT FROM AUTHOR]

In this paper, we give some remarks on the recent papers on generalized F-contractions. Our results unify and generalize the previous results in the existing literature. Moreover, we give an example to support our results. As an application, we give the existence and uniqueness of a solution to a class of differential equations. [ABSTRACT FROM AUTHOR]

This article has a motive to derive a new class of differential equations and associated integral equations for some hybrid families of Laguerre–Gould–Hopper-based Sheffer polynomials. We derive recurrence relations, differential equation, integro-differential equation, and integral equation for the Laguerre–Gould–Hopper-based Sheffer polynomials by using the factorization method. [ABSTRACT FROM AUTHOR]

Functional, integral and differential equations of transformed probability generating functions are generally recognized as powerful analytical tools for establishing characterizations of discrete probability distributions. The present paper establishes a characterization of the distribution of an important integral part model by incorporating an integral equation based on three fundamental transformed probability generating functions. Interpretations of such a characterization in analyzing and implementing information risk frequency reduction operations are also established. [ABSTRACT FROM AUTHOR]

In this paper, we generalize some integral inequalities, then we establish bounds on the solutions and show the usefulness of our results in investigating the stability on the solutions of integral equations, differential equations and integro-differential systems. [ABSTRACT FROM AUTHOR]

We investigate an integral equation with Bessel potential in ℝ n+ and prove that every positive solution of the integral equation is rotationally symmetric about some line parallel to the -axis. Furthermore, we consider an overdetermined problem of the same integral equation in a cylinder type domain and obtain the symmetry of both domains and solutions, where is a bounded domain with . [ABSTRACT FROM AUTHOR]

We investigate an integral equation with Bessel potential in ℝ n+ and prove that every positive solution of the integral equation is rotationally symmetric about some line parallel to the -axis. Furthermore, we consider an overdetermined problem of the same integral equation in a cylinder type domain and obtain the symmetry of both domains and solutions, where is a bounded domain with . [ABSTRACT FROM AUTHOR]

In this paper, we define and study a generalized multiparameter function of Mittag-Leffler type. Some special cases and basic properties including its Mellin–Barnes-type integral representation, recurrence relation, and expansion are given. The Laplace transform and the first Sonine integral of the generalized function are evaluated. Differential and integral equations with the Erdelyi–Kober operator are established. The Cauchy problem for the multiparameter differential equation is formulated and its solution is obtained. [ABSTRACT FROM PUBLISHER]

Let T be a time scale which is unbounded above and below and such that 0 ϵ T. Let id - r : T → T be such that (id - r) (T) is a time scale. We use the contraction mapping theorem to obtain stability results about the zero solution for the following neutral nonlinear dynamic equations with unbounded delay χ▵ (t) = -a(t) χσ (t) + b(t)G(χ²(t),χ²(t - r(t))) + c(t)(χ²) ▵ (t - r(t), t ϵ(t))) +c(t) (χ²)▵ (t - r(t)), t ϵ T, and χ▵ (t) = -a(t) χσ (t) + b(t)G(χ²(t),χ²(t - r(t))) + c(t)(χ²) ▵ (t - r(t), t ϵ(t))) +c(t) (χ²)▵ (t - r(t)), t ϵ T, where ƒ▵ is the ▵-derivative on T and ƒ▵ is the ▵-derivative on (id - r) (T) . We provide interesting examples to illustrate our claims. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the asymptotic stability of a generalized linear neutral differential equation with variable delays by using the fixed point theory. An asymptotic stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton (2003) , Zhang (2005) , Raffoul (2004) , and Jin and Luo (2008) . Two examples are also given to illustrate our results. [Copyright &y& Elsevier]

The quasi-Green's function method is used to solve the free vibration problem of clamped thin plates on the Winkler foundation. Quasi-Green's function is established by the fundamental solution and the boundary equation of the problem. The function satisfies the homogeneous boundary condition of the problem. The mode-shape differential equation of the free vibration problem of clamped thin plates on the Winkler foundation is reduced to the Fredholm integral equation of the second kind by Green's formula. The irregularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The numerical results show the high accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we generalize some integral inequalities to more general situations. We establish bounds on the solutions and, by means of examples, we show the usefulness of our results in investigating the global existence of the solutions of integral equations and differential equations. [Copyright &y& Elsevier]

In this article, I outline how integral equations and Green's functions are used in the courses that I teach, and how I believe my students benefit from exposure to these topics. [ABSTRACT FROM AUTHOR]

Abstract: Automatic Chebyshev spectral collocation methods for Fredholm and Volterra integral and integro-differential equations have been implemented as part of the chebfun software system. This system enables a symbolic syntax to be applied to numerical objects in order to pose and solve problems without explicit references to discretization. The same objects can be used in matrix-free iterative methods in linear algebra, in order to avoid very large dense matrices or allow application to problems with nonsmooth coefficients. As a further application of the ability to implement operator equations, a method of Greengard for the recasting of differential equations as integral equations is generalized to mth order boundary value and generalized eigenvalue problems. In the integral form, large condition numbers associated with differentiation matrices in high-order problems are avoided. The ability to implement the recasting process generally follows from implementation of the operator expressions in chebfun. The integral method also can be extended to first-order systems, although chebfun syntax does not currently allow easy implementation in this case. [Copyright &y& Elsevier]