Maha Belhadj, Jamal Rezaei Roshan, Mohamed Boumaiza, Vahid Parvaneh
{"title":"FIXED-POINT THEOREMS FOR MEIR–KEELER MULTIVALUED MAPS AND APPLICATION","authors":"Maha Belhadj, Jamal Rezaei Roshan, Mohamed Boumaiza, Vahid Parvaneh","doi":"10.1216/jie.2022.34.389","DOIUrl":"https://doi.org/10.1216/jie.2022.34.389","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49332845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"SOLVABILITY AND APPROXIMATION OF NONLINEAR FUNCTIONAL MIXED VOLTERRA–FREDHOLM EQUATION IN BANACH SPACE","authors":"C. Nwaigwe","doi":"10.1216/jie.2022.34.489","DOIUrl":"https://doi.org/10.1216/jie.2022.34.489","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43681476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"MIXED BOUNDARY VALUE PROBLEMS FOR THE HELMHOLTZ EQUATION","authors":"D. Natroshvili, T. Tsertsvadze","doi":"10.1216/jie.2022.34.475","DOIUrl":"https://doi.org/10.1216/jie.2022.34.475","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45595676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present article, we consider an integro-differential Dirac system with an integral delay on a finite interval. We obtain the asymptotical formula for the nodal points of the first components of the eigenfunctions, formulate a uniqueness theorem and prove that the kernel of the Dirac operator can be uniquely determined from a dense subset of the nodal set. We also present examples for reconstructing the kernel by using the nodal points.
{"title":"INVERSE NODAL PROBLEM FOR THE INTEGRODIFFERENTIAL DIRAC OPERATOR WITH A DELAY IN THE KERNEL","authors":"S. Mosazadeh","doi":"10.1216/jie.2022.34.465","DOIUrl":"https://doi.org/10.1216/jie.2022.34.465","url":null,"abstract":"In the present article, we consider an integro-differential Dirac system with an integral delay on a finite interval. We obtain the asymptotical formula for the nodal points of the first components of the eigenfunctions, formulate a uniqueness theorem and prove that the kernel of the Dirac operator can be uniquely determined from a dense subset of the nodal set. We also present examples for reconstructing the kernel by using the nodal points.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42376849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"CONTROLLABILITY OF SEMILINEAR NEUTRAL STOCHASTIC INTEGRODIFFERENTIAL EVOLUTION SYSTEMS WITH FRACTIONAL BROWNIAN MOTION","authors":"Nan Cao, Xianlong Fu","doi":"10.1216/jie.2022.34.409","DOIUrl":"https://doi.org/10.1216/jie.2022.34.409","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48886791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work addresses the regularity of solutions for a nonlocal diffusion equation over the space of periodic distributions. The spatial operator for the nonlocal diffusion equation is given by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a unified approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the study of regular as well as distributional solutions of the nonlocal diffusion equation, integrable as well as singular kernels, in any spatial dimension. In addition, the results extend beyond operators with singular kernels to nonlocal super-diffusion operators. We present results on the spatial and temporal regularity of solutions in terms of regularity of the initial data or the diffusion source term. Moreover, solutions of the nonlocal diffusion equation are shown to converge to the solution of the classical diffusion equation for two types of limits: as the spatial nonlocality vanishes or as the singularity of the integral kernel approaches a certain critical singularity that depends on the spatial dimension. Furthermore, we show that, for the case of integrable kernels, discontinuities in the initial data propagate and persist in the solution of the nonlocal diffusion equation. The magnitude of a jump discontinuity is shown to decay overtime.
{"title":"REGULARITY OF SOLUTIONS FOR NONLOCAL DIFFUSION EQUATIONS ON PERIODIC DISTRIBUTIONS","authors":"I. Mustapha, Bacim Alali, Nathan Albin","doi":"10.1216/jie.2023.35.81","DOIUrl":"https://doi.org/10.1216/jie.2023.35.81","url":null,"abstract":"This work addresses the regularity of solutions for a nonlocal diffusion equation over the space of periodic distributions. The spatial operator for the nonlocal diffusion equation is given by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a unified approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the study of regular as well as distributional solutions of the nonlocal diffusion equation, integrable as well as singular kernels, in any spatial dimension. In addition, the results extend beyond operators with singular kernels to nonlocal super-diffusion operators. We present results on the spatial and temporal regularity of solutions in terms of regularity of the initial data or the diffusion source term. Moreover, solutions of the nonlocal diffusion equation are shown to converge to the solution of the classical diffusion equation for two types of limits: as the spatial nonlocality vanishes or as the singularity of the integral kernel approaches a certain critical singularity that depends on the spatial dimension. Furthermore, we show that, for the case of integrable kernels, discontinuities in the initial data propagate and persist in the solution of the nonlocal diffusion equation. The magnitude of a jump discontinuity is shown to decay overtime.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43840927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate properties of a family of integral operators B with a weakly singular compactly supported zonal kernel function on the surface S of the unit 3D sphere. The support is over a spherical cap of height h ∈ (0,2]. Operators like this arise in some common types of approximations of time domain boundary integral equations (TDBIE) describing the scattering of acoustic waves from the surface of the sphere embedded in an infinite homogeneous medium where h is directly related to the time step size. We show that the Legendre polynomials of degree `≥ 0 satisfy ∫ h 0 P̀ (1−z2/2)dz > 0 for all h∈ (0,2] and, using spherical harmonics and the Funk-Hecke formula for the eigenvalues of B, that this is a key to unlocking positivity results for a subfamily of these operators. As well as positivity results we give detailed upper and lower bounds on the eigenvalues of B and on ∫ S u(x)(Bu)(x) dx. We give various examples of where these results are useful in numerical approximations of the TDBIE on the sphere and show that positivity of B is a necessary condition for these approximation schemes to be well-defined. We also show the connection between the results for eigenvalues and the separation of variables solution of the TDBIE on the sphere. Finally we show how this relates to scattering from an infinite flat surface and Cooke’s 1937 result ∫ r 0 J0(z)dz > 0 for all r > 0.
{"title":"Positivity of a weakly singular operator and approximation of wave scattering from the sphere","authors":"D. Duncan","doi":"10.1216/jie.2022.34.317","DOIUrl":"https://doi.org/10.1216/jie.2022.34.317","url":null,"abstract":"We investigate properties of a family of integral operators B with a weakly singular compactly supported zonal kernel function on the surface S of the unit 3D sphere. The support is over a spherical cap of height h ∈ (0,2]. Operators like this arise in some common types of approximations of time domain boundary integral equations (TDBIE) describing the scattering of acoustic waves from the surface of the sphere embedded in an infinite homogeneous medium where h is directly related to the time step size. We show that the Legendre polynomials of degree `≥ 0 satisfy ∫ h 0 P̀ (1−z2/2)dz > 0 for all h∈ (0,2] and, using spherical harmonics and the Funk-Hecke formula for the eigenvalues of B, that this is a key to unlocking positivity results for a subfamily of these operators. As well as positivity results we give detailed upper and lower bounds on the eigenvalues of B and on ∫ S u(x)(Bu)(x) dx. We give various examples of where these results are useful in numerical approximations of the TDBIE on the sphere and show that positivity of B is a necessary condition for these approximation schemes to be well-defined. We also show the connection between the results for eigenvalues and the separation of variables solution of the TDBIE on the sphere. Finally we show how this relates to scattering from an infinite flat surface and Cooke’s 1937 result ∫ r 0 J0(z)dz > 0 for all r > 0.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41696948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a numerical algorithm via a modified hat functions (MHFs) has been proposed to solve a class of non-linear fractional Volterra integral equations of the second kind. A fractional-order operational matrix of integration is introduced. In a new methodology, the operational matrices of MHFs and the powers of weakly singular kernels of integral equations are used as a structure for transforming the main problem into a number of systems consisting of two equations for two unknowns. Relative errors for the approximated solutions are investigated. Convergence analysis of the proposed method is evaluated and convergence rate is addressed. Part ultimate, the extraordinary accuracy of the utilized approach is illustrated by a few examples. The results, absolute and relative errors are illustrated in some Tables and diagrams. In addition, a comparison is made between the absolute errors obtained by the proposed method and two other methods; one using a hybrid approach and the other applies second Chebyshev wavelet.
{"title":"A numerical algorithm for a class of nonlinear fractional Volterra integral equations via modified hat functions","authors":"J. Biazar, H. Ebrahimi","doi":"10.1216/jie.2022.34.295","DOIUrl":"https://doi.org/10.1216/jie.2022.34.295","url":null,"abstract":"In this paper, a numerical algorithm via a modified hat functions (MHFs) has been proposed to solve a class of non-linear fractional Volterra integral equations of the second kind. A fractional-order operational matrix of integration is introduced. In a new methodology, the operational matrices of MHFs and the powers of weakly singular kernels of integral equations are used as a structure for transforming the main problem into a number of systems consisting of two equations for two unknowns. Relative errors for the approximated solutions are investigated. Convergence analysis of the proposed method is evaluated and convergence rate is addressed. Part ultimate, the extraordinary accuracy of the utilized approach is illustrated by a few examples. The results, absolute and relative errors are illustrated in some Tables and diagrams. In addition, a comparison is made between the absolute errors obtained by the proposed method and two other methods; one using a hybrid approach and the other applies second Chebyshev wavelet.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42203783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the asymptotic behavior of a viscoelastic neutral di erential equation. A stability with an explicit decay result of the energy associated to the problem is established. It is found that the energy decay rate is optimal, in the sense that, it is the same as that of the relaxation function.
{"title":"Optimal stability for a viscoelastic neutral differential problem","authors":"J. Hassan, N. Tatar","doi":"10.1216/jie.2022.34.335","DOIUrl":"https://doi.org/10.1216/jie.2022.34.335","url":null,"abstract":"We investigate the asymptotic behavior of a viscoelastic neutral di erential equation. A stability with an explicit decay result of the energy associated to the problem is established. It is found that the energy decay rate is optimal, in the sense that, it is the same as that of the relaxation function.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42935439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we are interested in the convergence of a system of integro-differential equations with respect to an asymptotic parameter ε. It appears in the context of cell adhesion modelling [16, 15]. We extend the framework from [12, 13], strongly depending on the hypothesis that the external load f is in Lip([0, T ]) to the case where f ∈ BV(0, T ) only. We show how results presented in [13] naturally extend to this new setting, while only partial results can be obtained following the comparison principle introduced in [12].
{"title":"Friction mediated by transient elastic linkages: extension to loads of bounded variation","authors":"S. Allouch, V. Milišić","doi":"10.1216/jie.2022.34.267","DOIUrl":"https://doi.org/10.1216/jie.2022.34.267","url":null,"abstract":"In this work, we are interested in the convergence of a system of integro-differential equations with respect to an asymptotic parameter ε. It appears in the context of cell adhesion modelling [16, 15]. We extend the framework from [12, 13], strongly depending on the hypothesis that the external load f is in Lip([0, T ]) to the case where f ∈ BV(0, T ) only. We show how results presented in [13] naturally extend to this new setting, while only partial results can be obtained following the comparison principle introduced in [12].","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49517034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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