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13 pages, 2982 KB

Open AccessArticle

Stability Analysis of the Solution for the Mixed Integral Equation with Symmetric Kernel in Position and Time with Its Applications

by Faizah M. Alharbi

Symmetry 2024, 16(8), 1048; https://doi.org/10.3390/sym16081048 - 14 Aug 2024

Viewed by 875

Abstract

Under certain assumptions, the existence of a unique solution of mixed integral equation (MIE) of the second type with a symmetric kernel is discussed, in

L_{2} [Ω] \times C [0, T], T < 1,

Ω

is the [...] Read more.

Under certain assumptions, the existence of a unique solution of mixed integral equation (MIE) of the second type with a symmetric kernel is discussed, in

L_{2} [Ω] \times C [0, T], T < 1,

Ω

is the position domain of integration and T is the time. The convergence error and the stability error are considered. Then, after using the separation technique, the MIE transforms into a system of Hammerstein integral equations (SHIEs) with time-varying coefficients. The nonlinear algebraic system (NAS) is obtained after using the degenerate method. New and special cases are derived from this work. Moreover, numerical results are computed using MATLAB R2023a software. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Equations: Mathematical Models, Methods and Applications Ⅱ)

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13 pages, 269 KB

Open AccessArticle

New Results on Ulam Stabilities of Nonlinear Integral Equations

by Osman Tunç, Cemil Tunç and Jen-Chih Yao

Mathematics 2024, 12(5), 682; https://doi.org/10.3390/math12050682 - 26 Feb 2024

Cited by 13 | Viewed by 1398

Abstract

This article deals with the study of Hyers–Ulam stability (HU stability) and Hyers–Ulam–Rassias stability (HUR stability) for two classes of nonlinear Volterra integral equations (VIEqs), which are Hammerstein-type integral and Hammerstein-type functional integral equations, respectively. In this article, both the HU stability and [...] Read more.

This article deals with the study of Hyers–Ulam stability (HU stability) and Hyers–Ulam–Rassias stability (HUR stability) for two classes of nonlinear Volterra integral equations (VIEqs), which are Hammerstein-type integral and Hammerstein-type functional integral equations, respectively. In this article, both the HU stability and HUR stability are obtained for the first integral equation and the HUR stability is obtained for the second integral equation. Among the used techniques, we present fixed point arguments and the Gronwall lemma as a basic tool. Two supporting examples are also provided to demonstrate the applications and effectiveness of the results. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

20 pages, 4222 KB

Open AccessArticle

A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a Quadrature Nystrom Method

by A. R. Jan, M. A. Abdou and M. Basseem

Fractal Fract. 2023, 7(9), 656; https://doi.org/10.3390/fractalfract7090656 - 31 Aug 2023

Cited by 5 | Viewed by 1506

Abstract

In this work, the existence and uniqueness solution of the fractional nonlinear mixed integro-differential equation (FrNMIoDE) is guaranteed with a general discontinuous kernel based on position and time-space [...] Read more.

In this work, the existence and uniqueness solution of the fractional nonlinear mixed integro-differential equation (FrNMIoDE) is guaranteed with a general discontinuous kernel based on position and time-space

L_{2} [Ω] \times C [0, T], T < 1

. The FrNMIoDE conformed to the Volterra-Hammerstein integral equation (V-HIE) of the second kind, after applying the characteristics of a fractional integral, with a general discontinuous kernel in position for the Hammerstein integral term and a continuous kernel in time to the Volterra integral (VI) term. Then, using a separation technique methodology, we developed HIE, whose physical coefficients were time-variable. By examining the system’s convergence, the product Nystrom technique (PNT) and associated schemes were employed to create a nonlinear algebraic system (NAS). Full article

(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)

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20 pages, 318 KB

Open AccessArticle

Solving Integral Equations via Fixed Point Results Involving Rational-Type Inequalities

by Syed Shah Khayyam, Muhammad Sarwar, Asad Khan, Nabil Mlaiki and Fatima M. Azmi

Axioms 2023, 12(7), 685; https://doi.org/10.3390/axioms12070685 - 12 Jul 2023

Cited by 6 | Viewed by 1915

Abstract

In this study, we establish unique and common fixed point results in the context of a complete complex-valued b-metric space using rational-type inequalities. The presented work generalizes some well-known results from the existing literature. Furthermore, to ensure the validity of the findings, we [...] Read more.

In this study, we establish unique and common fixed point results in the context of a complete complex-valued b-metric space using rational-type inequalities. The presented work generalizes some well-known results from the existing literature. Furthermore, to ensure the validity of the findings, we have included some examples and a section on the existence of solutions for the systems of Volterra–Hammerstein integral equations and Urysohn integral equations, respectively. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)

19 pages, 659 KB

Open AccessArticle

A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a General Discontinuous Kernel

by Sharifah E. Alhazmi and Mohamed A. Abdou

Fractal Fract. 2023, 7(2), 173; https://doi.org/10.3390/fractalfract7020173 - 9 Feb 2023

Cited by 9 | Viewed by 1961

Abstract

In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness of the solution is provided through the principal form of the integral equation, [...] Read more.

In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness of the solution is provided through the principal form of the integral equation, based on the Banach fixed point theorem. After applying the properties of a fractional integral, the Fr-NMIDE conformed to the Volterra–Hammerstein integral equation (V-HIE) of the second kind, with a general discontinuous kernel in position with the Hammerstein integral term and a continuous kernel in time to the Volterra term. Then, using a technique of the separating method, we obtained HIE, where its physical coefficients were variable in time. The Toeplitz matrix method (TMM) and its schemes were used to obtain a nonlinear algebraic system by studying the convergence of the system. The Maple 18 program was implemented to present the numerical results, along with corresponding errors. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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16 pages, 309 KB

Open AccessArticle

A Symbolic Method for Solving a Class of Convolution-Type Volterra–Fredholm–Hammerstein Integro-Differential Equations under Nonlocal Boundary Conditions

by Efthimios Providas and Ioannis Nestorios Parasidis

Algorithms 2023, 16(1), 36; https://doi.org/10.3390/a16010036 - 7 Jan 2023

Viewed by 1931

Abstract

Integro-differential equations involving Volterra and Fredholm operators (VFIDEs) are used to model many phenomena in science and engineering. Nonlocal boundary conditions are more effective, and in some cases necessary, because they are more accurate measurements of the true state than classical (local) initial [...] Read more.

Integro-differential equations involving Volterra and Fredholm operators (VFIDEs) are used to model many phenomena in science and engineering. Nonlocal boundary conditions are more effective, and in some cases necessary, because they are more accurate measurements of the true state than classical (local) initial and boundary conditions. Closed-form solutions are always desirable, not only because they are more efficient, but also because they can be valuable benchmarks for validating approximate and numerical procedures. This paper presents a direct operator method for solving, in closed form, a class of Volterra–Fredholm–Hammerstein-type integro-differential equations under nonlocal boundary conditions when the inverse operator of the associated Volterra integro-differential operator exists and can be found explicitly. A technique for constructing inverse operators of convolution-type Volterra integro-differential operators (VIDEs) under multipoint and integral conditions is provided. The proposed methods are suitable for integration into any computer algebra system. Several linear and nonlinear examples are solved to demonstrate the effectiveness of the method. Full article

(This article belongs to the Special Issue Computational Methods and Optimization for Numerical Analysis)

13 pages, 357 KB

Open AccessArticle

A Collocation Method for Mixed Volterra–Fredholm Integral Equations of the Hammerstein Type

by Sanda Micula

Mathematics 2022, 10(17), 3044; https://doi.org/10.3390/math10173044 - 23 Aug 2022

Cited by 1 | Viewed by 1752

Abstract

This paper presents a collocation method for the approximate solution of two-dimensional mixed Volterra–Fredholm integral equations of the Hammerstein type. For a reformulation of the equation, we consider the domain of integration as a planar triangle and use a special type of linear [...] Read more.

This paper presents a collocation method for the approximate solution of two-dimensional mixed Volterra–Fredholm integral equations of the Hammerstein type. For a reformulation of the equation, we consider the domain of integration as a planar triangle and use a special type of linear interpolation on triangles. The resulting quadrature formula has a higher degree of precision than expected, leading to a collocation method that is superconvergent at the collocation nodes. The convergence of the method is established, as well as the rate of convergence. Numerical examples are considered, showing the applicability of the proposed scheme and the agreement with the theoretical results. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)

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9 pages, 248 KB

Open AccessArticle

On a Functional Integral Equation

by Daniela Marian, Sorina Anamaria Ciplea and Nicolaie Lungu

Symmetry 2021, 13(8), 1321; https://doi.org/10.3390/sym13081321 - 22 Jul 2021

Cited by 12 | Viewed by 2089

Abstract

In this paper, we establish some results for a Volterra–Hammerstein integral equation with modified arguments: existence and uniqueness, integral inequalities, monotony and Ulam-Hyers-Rassias stability. We emphasize that many problems from the domain of symmetry are modeled by differential and integral equations and those [...] Read more.

In this paper, we establish some results for a Volterra–Hammerstein integral equation with modified arguments: existence and uniqueness, integral inequalities, monotony and Ulam-Hyers-Rassias stability. We emphasize that many problems from the domain of symmetry are modeled by differential and integral equations and those are approached in the stability point of view. In the literature, Fredholm, Volterra and Hammerstein integrals equations with symmetric kernels are studied. Our results can be applied as particular cases to these equations. Full article

(This article belongs to the Special Issue Ulam's Type Stability and Symmetries)

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