Relevant bibliographies by topics / Nonlocal equation

Academic literature on the topic 'Nonlocal equation'

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Published: 9 March 2023
Last updated: 10 March 2023

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Journal articles on the topic "Nonlocal equation"

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Lukashchuk, Stanislav Yu. "Approximate Nonlocal Symmetries for a Perturbed Schrödinger Equation with a Weak Infinite Power-Law Memory." AppliedMath 2, no. 4 (October 17, 2022): 585–608. http://dx.doi.org/10.3390/appliedmath2040034.

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A nonlocally perturbed linear Schrödinger equation with a small parameter was derived under the assumption of low-level fractionality by using one of the known general nonlocal wave equations with an infinite power-law memory. The problem of finding approximate symmetries for the equation is studied here. It has been shown that the perturbed Schrödinger equation inherits all symmetries of the classical linear equation. It has also been proven that approximate symmetries corresponding to Galilean transformations and projective transformations of the unperturbed equation are nonlocal. In addition, a special class of nonlinear, nonlocally perturbed Schrödinger equations that admits an approximate nonlocal extension of the Galilei group is derived. An example of constructing an approximately invariant solution for the linear equation using approximate scaling symmetry is presented.
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Peng, Linyu. "Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations." Symmetry 11, no. 7 (July 5, 2019): 884. http://dx.doi.org/10.3390/sym11070884.

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In this paper, symmetry analysis is extended to study nonlocal differential equations. In particular, two integrable nonlocal equations are investigated, the nonlocal nonlinear Schrödinger equation and the nonlocal modified Korteweg–de Vries equation. Based on general theory, Lie point symmetries are obtained and used to reduce these equations to nonlocal and local ordinary differential equations, separately; namely, one symmetry may allow reductions to both nonlocal and local equations, depending on how the invariant variables are chosen. For the nonlocal modified Korteweg–de Vries equation, analogously to the local situation, all reduced local equations are integrable. We also define complex transformations to connect nonlocal differential equations and differential-difference equations.
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Feng, Wei, and Song-Lin Zhao. "Soliton solutions to the nonlocal non-isospectral nonlinear Schrödinger equation." International Journal of Modern Physics B 34, no. 25 (September 9, 2020): 2050219. http://dx.doi.org/10.1142/s0217979220502197.

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In this paper we study the nonlocal reductions for the non-isospectral Ablowitz-Kaup-Newell-Segur equation. By imposing the real and complex nonlocal reductions on the non-isospectral Ablowitz-Kaup-Newell-Segur equation, we derive two types of nonlocal non-isospectral nonlinear Schrödinger equations, in which one is real nonlocal non-isospectral nonlinear Schrödinger equation and the other is complex nonlocal non-isospectral nonlinear Schrödinger equation. Of both of these two equations, there are the reverse time nonlocal type and the reverse space nonlocal type. Soliton solutions in terms of double Wronskian to the reduced equations are obtained by imposing constraint conditions on the double Wronskian solutions of the non-isospectral Ablowitz-Kaup-Newell-Segur equation. Dynamics of the one-soliton solutions are analyzed and illustrated by asymptotic analysis.
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Khare, Avinash, and Avadh Saxena. "Novel superposed kinklike and pulselike solutions for several nonlocal nonlinear equations." Journal of Mathematical Physics 63, no. 12 (December 1, 2022): 122903. http://dx.doi.org/10.1063/5.0109384.

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We show that a number of nonlocal nonlinear equations, including the Ablowitz–Musslimani and Yang variant of the nonlocal nonlinear Schrödinger (NLS) equation, the nonlocal modified Korteweg de Vries (mKdV) equation, and the nonlocal Hirota equation, admit novel kinklike and pulselike superposed periodic solutions. Furthermore, we show that the nonlocal mKdV equation also admits the superposed (hyperbolic) kink–antikink solution. In addition, we show that while the nonlocal Ablowitz–Musslimani variant of the NLS admits complex parity-time reversal-invariant kink and pulse solutions, neither the local NLS nor the Yang variant of the nonlocal NLS admits such solutions. Finally, except for the Yang variant of the nonlocal NLS, we show that the other three nonlocal equations admit both the kink and pulse solutions in the same model.
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Xu, Hai Jing, and Song Lin Zhao. "Local and Nonlocal Reductions of Two Nonisospectral Ablowitz-Kaup-Newell-Segur Equations and Solutions." Symmetry 13, no. 1 (December 24, 2020): 23. http://dx.doi.org/10.3390/sym13010023.

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In this paper, local and nonlocal reductions of two nonisospectral Ablowitz-Kaup-Newell-Segur equations, the third order nonisospectral AKNS equation and the negative order nonisospectral AKNS equation, are studied. By imposing constraint conditions on the double Wronskian solutions of the aforesaid nonisospectral AKNS equations, various solutions for the local and nonlocal nonisospectral modified Korteweg-de Vries equation and local and nonlocal nonisospectral sine-Gordon equation are derived, including soliton solutions and Jordan block solutions. Dynamics of some obtained solutions are analyzed and illustrated by asymptotic analysis.
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Delgado, M., A. Suárez, and I. B. M. Duarte. "Nonlocal problems arising from the birth-jump processes." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 149, no. 2 (December 27, 2018): 447–69. http://dx.doi.org/10.1017/prm.2018.34.

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In this paper, we prove the existence and uniqueness of a positive solution for a nonlocal logistic equation arising from the birth-jump processes. For this, we establish a sub-super solution method for nonlocal elliptic equations, we perform a study of the eigenvalue problems associated with these equations and we apply these results to the nonlocal logistic equation.
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Feng, Wei, Song-Lin Zhao, and Ying-Ying Sun. "Double Casoratian solutions to the nonlocal semi-discrete modified Korteweg-de Vries equation." International Journal of Modern Physics B 34, no. 05 (February 3, 2020): 2050021. http://dx.doi.org/10.1142/s0217979220500216.

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Two nonlocal versions of the semi-discrete modified Korteweg-de Vries equation are derived by different nonlocal reductions from a coupled equation set in the Ablowitz–Ladik hierarchy. Different kinds of exact solutions in terms of double Casoratians to the reduced equations are obtained by imposing constraint conditions on the double Casorati determinant solutions of the coupled equation set. Dynamics of the soliton solutions for the real and complex nonlocal semi-discrete modified Korteweg-de Vries equations are analyzed and illustrated by asymptotic analysis.
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Gaygusuzoglu, Guler, Metin Aydogdu, and Ufuk Gul. "Nonlinear Wave Modulation in Nanorods Using Nonlocal Elasticity Theory." International Journal of Nonlinear Sciences and Numerical Simulation 19, no. 7-8 (December 19, 2018): 709–19. http://dx.doi.org/10.1515/ijnsns-2017-0225.

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AbstractIn this study, nonlinear wave modulation in nanorods is examined on the basis of nonlocal elasticity theory. Eringen's nonlocal elasticity theory is employed to derive nonlinear equations for the motion of nanorods. The analysis of the modulation of axial waves in nonlocal elastic media is performed, and the reductive perturbation method is used for the solution of the nonlinear equations. The propagation of weakly nonlinear and strongly dispersive waves is investigated, and the nonlinear Schrödinger (NLS) equation is acquired as an evolution equation. For the purpose of a numerical investigation of the nonlocal impacts on the NLS equation, it has been investigated whether envelope solitary wave solutions exist by utilizing the physical and geometric features of the carbon nanotubes. Amplitude dependent wave frequencies, phase and group velocities have been obtained and they have compared for the linear local, the linear nonlocal, the nonlinear local and the nonlinear nonlocal cases.
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Kulagin, Anton E., Alexander V. Shapovalov, and Andrey Y. Trifonov. "Semiclassical Spectral Series Localized on a Curve for the Gross–Pitaevskii Equation with a Nonlocal Interaction." Symmetry 13, no. 7 (July 17, 2021): 1289. http://dx.doi.org/10.3390/sym13071289.

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We propose the approach to constructing semiclassical spectral series for the generalized multidimensional stationary Gross–Pitaevskii equation with a nonlocal interaction term. The eigenvalues and eigenfunctions semiclassically concentrated on a curve are obtained. The curve is described by the dynamic system of moments of solutions to the nonlocal Gross–Pitaevskii equation. We solve the eigenvalue problem for the nonlocal stationary Gross–Pitaevskii equation basing on the semiclassical asymptotics found for the Cauchy problem of the parametric family of linear equations associated with the time-dependent Gross–Pitaevskii equation in the space of extended dimension. The approach proposed uses symmetries of equations in the space of extended dimension.
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Wang, Xiaodong, Jianping Wu, Yazi Wang, and Can Chen. "Extended Tanh-Function Method and Its Applications in Nonlocal Complex mKdV Equations." Mathematics 10, no. 18 (September 7, 2022): 3250. http://dx.doi.org/10.3390/math10183250.

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In order to construct the multiple traveling wave solutions of the nonlocal modified Korteweg de Vires (mKdV) equation, the modified tanh-function approach for local soliton equations is extended to a nonlocal complex mKdV equation. The central idea of this method is to use the solution of the Riccati equation to replace the tanh function in the tanh function (THF) method. As an application, we investigate a new traveling wave solution for the nonlocal complex mKdV equation of Ablowitz and Musslimani. Moreover, some exciting diagrams show the underlying dynamics of some given solutions.
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Dissertations / Theses on the topic "Nonlocal equation"

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Alshomrani, Ali. "Analysis and approximation of a nonlocal equation." Thesis, Heriot-Watt University, 2014. http://hdl.handle.net/10399/2769.

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Khan, K. B. "The nonlocal-nonlinear-Schroedinger-equation model of superfluid '4He." Thesis, University of Exeter, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.267224.

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Jachimavičienė, Justina. "Solution of a pseudoparabolic equation with nonlocal integral conditions by the finite difference method." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2013. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2013~D_20130220_160906-66807.

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The thesis analyzes the third-order one-dimensional pseudoparabolic equations with two types of nonlocal conditions. The stability of difference schemes for this problem was studied using the analysis of the spectrum structure of a difference operator with nonlocal conditions. The analysis of the increased accuracy difference schemes for third-order one-dimensional and two-dimensional pseudoparabolic equations with integral conditions has been made. The thesis considers a two-dimensional pseudoparabolic equation with nonlocal integral conditions in one coordinate direction. This problem was solved by a locally one-dimensional method. The stability of a difference scheme has been investigated based on the spectrum structure. The doctoral disertation investigates three-layer difference schemes for one-dimensional pseudoparabolic equations with various, including nonlocal, conditions. Also, the conditions for the stability of three-layer explicit difference schemes have been explored.
Disertacijoje išnagrinėta trečiosios eilės vienmatė pseudoparabolinė lygtis su dviejų tipų nelokaliosiomis sąlygomis. Šiems uždaviniams spręsti sudarytos skirtuminės schemos, kurių stabilumas tiriamas, taikant skirtuminių operatorių su nelokaliosiomis sąlygomis spektro struktūrą. Trečiosios eilės vienmatėms ir dvimatėms pseudoparabolinėms lygtims su integralinėmis sąlygomis sudarytos ir išnagrinėtos padidinto tikslumo skirtuminės schemos. Išnagrinėta dvimatė pseudoparabolinė lygtis su nelokaliosiomis integralinėmis sąlygomis viena koordinačių kryptimi. Tokiam uždaviniui spręsti pritaikytas ir išnagrinėtas lokaliai vienmatis metodas, ištirtos šio metodo stabilumo sąlygos. Taip pat išnagrinėtos: trisluoksnės skirtuminės schemos vienmatei pseudoparabolinei lygčiai su įvairiomis, taip pat ir nelokaliosiomis, sąlygomis; trisluoksnių išreikštinių skirtuminių schemų stabilumo sąlygos.
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BRASSEUR, JULIEN. "ANALYSIS OF SOME NONLOCAL MODELS IN POPULATION DYNAMICS." Doctoral thesis, Università degli Studi di Milano, 2018. http://hdl.handle.net/2434/597755.

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This thesis is mainly devoted to the mathematical analysis of some nonlocal models arising in population dynamics. In general, the study of these models meets with numerous difficulties owing to the lack of compactness and of regularizing effects. In this respect, their analysis requires new tools, both theoretical and qualitative. We present several results in this direction. In the first part, we develop a functional analytic toolbox which allows one to handle some quantities arising in the study of these models. In the first place, we extend the characterization of Sobolev spaces due to Bourgain, Brezis and Mironescu to low regularity function spaces of Besov type. This results in a new theoretical framework that is more adapted to the study of some nonlocal equations of Fisher-KPP type. In the second place, we study the regularity of the restrictions of these functions to hyperplanes. We prove that, for a large class of Besov spaces, a surprising loss of regularity occurs. Moreover, we obtain an optimal characterization of the regularity of these restrictions in terms of spaces of so-called “generalized smoothness”. In the second part, we study qualitative properties of solutions to some nonlocal reaction-diffusion equations set in (possibly) heterogeneous domains. In collaboration with J. Coville, F. Hamel and E. Valdinoci, we consider the case of a perforated domain which consists of the Euclidean space to which a compact set, called an “obstacle”, is removed. When the latter is convex (or close to being convex), we prove that the solutions are necessarily constant. In a joint work with J. Coville, we study in greater detail the influence of the geometry of the obstacle on the classification of the solutions. Using tools of the type of those developed in the first part of this thesis, we construct a family of counterexamples when the obstacle is no longer convex. Lastly, in a work in collaboration with S. Dipierro, we study qualitative properties of solutions to nonlinear elliptic systems in variational form. We establish various monotonicity results in a fairly general setting that covers both local and fractional operators.
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Roman, Svetlana. "Green's functions for boundary-value problems with nonlocal boundary conditions." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2011. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2011~D_20111227_092148-01085.

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In the dissertation, second-order and higher-order differential and discrete equations with additional conditions which are described by linearly independent linear functionals are investigated. The solutions to these problems, formulae and the existence conditions of Green's functions are presented, if the general solution of a homogeneous equation is known. The relation between two Green's functions of two nonhomogeneous problems for the same equation but with different additional conditions is obtained. These results are applied to problems with nonlocal boundary conditions. In the introduction the topicality of the problem is defined, the goals and tasks of the research are formulated, the scientific novelty of the dissertation, the methodology of research, the practical value and the significance of the results are presented. m-order differential problem and its Green's function are investigated in the first chapter. The relation between two Green's functions and the existence condition of Green's function are obtained. In the second chapter, the main definitions and results of the first chapter are formulated for the second-order differential equation with additional conditions. In the examples the application of the received results is analyzed for problems with nonlocal boundary conditions in detail. In the third chapter, the second-order difference equation with two additional conditions is considered. The expression of Green's function and its existence... [to full text]
Disertacijoje tiriami antros ir aukštesnės eilės diferencialinis ir diskretusis uždaviniai su įvairiomis, tame tarpe ir nelokaliosiomis, sąlygomis, kurios yra aprašytos tiesiškai nepriklausomais tiesiniais funkcionalais. Pateikiamos šių uždavinių Gryno funkcijų išraiškos ir jų egzistavimo sąlygos, jei žinoma homogeninės lygties fundamentalioji sistema. Gautas dviejų Gryno funkcijų sąryšis uždaviniams su ta pačia lygtimi, bet su papildomomis sąlygomis. Rezultatai pritaikomi uždaviniams su nelokaliosiomis kraštinėmis sąlygomis. Įvadiniame skyriuje aprašyta tiriamoji problema ir temos objektas, išanalizuotas temos aktualumas, išdėstyti darbo tikslai, uždaviniai, naudojama tyrimų metodika, mokslinis darbo naujumas ir gautų rezultatų reikšmė, pateikti ginamieji teiginiai ir darbo rezultatų aprobavimas. m-tosios eilės diferencialinis uždavinys ir jo Gryno funkcija nagrinėjami pirmajame skyriuje. Surastas uždavinio sprendinys, išreikštas per Gryno funkciją. Pateikta Gryno funkcijos egzistavimo sąlyga. Antrajame skyriuje pateikti pirmojo skyriaus pagrindiniai apibrėžimai ir rezultatai antros eilės diferencialinei lygčiai. Pavyzdžiuose išsamiai išanalizuotas gautų rezultatų pritaikymas uždaviniams su nelokaliosiomis kraštinėmis sąlygomis. Trečiajame skyriuje nagrinėjama antros eilės diskrečioji lygtis su dviem sąlygomis. Surastos diskrečiosios Gryno funkcijos išraiška ir jos egzistavimo sąlyga. Taip pat pateiktas dviejų Gryno funkcijų sąryšis, kuris leidžia surasti diskrečiosios... [toliau žr. visą tekstą]
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Fu, Xiaoming. "Reaction-diffusion Equations with Nonlinear and Nonlocal Advection Applied to Cell Co-culture." Thesis, Bordeaux, 2019. http://www.theses.fr/2019BORD0216/document.

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Cette thèse est consacrée à l’étude d’une classe d’équations de réaction-diffusion avec advection non-locale. La motivation vient du mouvement cellulaire avec le phénomène de ségrégation observé dans des expérimentations de co-culture cellulaire. La première partie de la thèse développe principalement le cadre théorique de notre modèle, à savoir le caractère bien posé du problème et le comportement asymptotique des solutions dans les cas d'une ou plusieurs espèces.Dans le Chapitre 1, nous montrons qu'une équation scalaire avec un noyau non-local ayant la forme d'une fonction étagée, peut induire des bifurcations de Turing et de Turing-Hopf avec le nombre d’ondes dominant aussi grand que souhaité. Nous montrons que les propriétés de bifurcation de l'état stable homogène sont intimement liées aux coefficients de Fourier du noyau non-local.Dans le Chapitre 2, nous étudions un modèle d'advection non-local à deux espèces avec inhibition de contact lorsque la viscosité est égale à zéro. En employant la notion de solution intégrée le long des caractéristiques, nous pouvons rigoureusement démontrer le caractère bien posé du problème ainsi que la propriété de ségrégation d'un tel système. Par ailleurs, dans le cadre de la théorie des mesures de Young, nous étudions le comportement asymptotique des solutions. D'un point de vue numérique, nous constatons que sous l'effet de la ségrégation, le modèle d'advection non-locale admet un principe d'exclusion.Dans le dernier Chapitre de la thèse, nous nous intéressons à l'application de nos modèles aux expérimentations de co-culture cellulaire. Pour cela, nous choisissons un modèle hyperbolique de Keller-Segel sur un domaine borné. En utilisant les données expérimentales, nous simulons un processus de croissance cellulaire durant 6 jours dans une boîte de pétri circulaire et nous discutons de l’impact de la propriété de ségrégation et des distributions initiales sur les proportions de la population finale
This thesis is devoted to the study for a class of reaction-diffusion equations with nonlocal advection. The motivation comes from the cell movement with segregation phenomenon observed in cell co-culture experiments. The first part of the thesis mainly develops the theoretical framework of our model, namely the well-posedness and asymptotic behavior of solutions in both single-species and multi-species cases.In Chapter 1, we show a single scalar equation with a step function kernel may display Turing and Turing-Hopf bifurcations with the dominant wavenumber as large as we want. We find the bifurcation properties of the homogeneous steady state is closed related to the Fourier coefficients of the nonlocal kernel.In Chapter 2, we study a two-species nonlocal advection model with contact inhibition when the viscosity equals zero. By employing the notion of the solution integrated along the characteristics, we rigorously prove the well-posedness and segregation property of such a hyperbolic nonlocal advection system. Besides, under the framework of Young measure theory, we investigate the asymptotic behavior of solutions. From a numerical perspective, we find that under the effect of segregation, the nonlocal advection model admits a competitive exclusion principle.In the last Chapter, we are interested in applying our models to a cell co-culturing experiment. To that aim, we choose a hyperbolic Keller-Segel model on a bounded domain. By utilizing the experimental data, we simulate a 6-day process of cell growth in a circular petri dish and discuss the impact of both the segregation property and initial distributions on the finial population proportions
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Cozzi, M. "QUALITATIVE PROPERTIES OF SOLUTIONS OF NONLINEAR ANISOTROPIC PDES IN LOCAL AND NONLOCAL SETTINGS." Doctoral thesis, Università degli Studi di Milano, 2016. http://hdl.handle.net/2434/345873.

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La tesi è dedicata allo studio di varie proprietà qualitative possedute dalle soluzioni di equazioni ellittiche poste nello spazio euclideo. L'attenzione principale del lavoro è rivolta a soluzioni intere di equazioni anisotrope/eterogenee che mostrano qualche genere di proprietà di simmetria e, in particolare, che posseggono simmetria unidimensionale. L'elaborato è diviso in due parti. La prima parte è riservata ad equazioni alle derivate parziali locali, mentre la seconda si concentra su di una classe meno usuale di equazioni non locali, determinate da operatori integrali.
The thesis is concerned with the study of several qualitative properties shared by the solutions of elliptic equations set in the Euclidean space. The main focus of the work is on entire solutions of anisotropic/heterogeneous equations that show some kind of symmetric properties and, in particular, that possess one-dimensional symmetry. The dissertation is divided into two parts. The first part deals with local partial differential equations, while the second one addresses a class of less familiar nonlocal equations driven by integral operators.
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DE, LUCA ALESSANDRA. "On some nonlocal issues: unique continuation from the boundary and capillarity problems for anisotropic kernels." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2022. http://hdl.handle.net/10281/378950.

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Lo scopo della presente tesi è quello di discutere i risultati ottenuti durante i miei studi di dottorato, principalmente rivolti a problemi non locali. Per prima cosa ci occupiamo di principi di continuazione unica forte ed espansioni asintotiche locali in determinati punti del bordo per soluzioni di due diverse classi di equazioni ellittiche. In particolare, partiamo con lo studio di una classe di equazioni ellittiche frazionarie in un dominio limitato sotto una condizione al contorno di Dirichlet omogenea esterna. Per fare ciò, sfruttiamo la procedura di estensione di Caffarelli-Silvestre, grazie alla quale il problema non locale può essere riformulato in modo equivalente come problema locale in una dimensione in più, generando un problema con condizioni miste. Dopodichè, utilizziamo un'idea classica di Garofalo e Lin per ottenere una condizione di raddoppio tramite una formula di monotonia per la funzione di Almgren. Per superare le difficoltà legate alla perdita di regolarità in corrispondenza della transizione tra le regioni di Dirichlet e di Neumann, introduciamo una nuova tecnica basata su un argomento di approssimazione, che ci permette di derivare una cosiddetta identità di tipo Pohozaev necessaria per stimare la derivata della funzione di Almgren . Otteniamo così un risultato di continuazione unica forte nel contesto locale, che viene a sua volta combinato con l’analisi di blow-up per dedurre espansioni asintotiche locali e, di conseguenza, una continuazione unica forte anche nel contesto non locale. Inoltre forniamo anche un risultato di continuazione unica forte dal bordo di una fessura per le soluzioni di una classe specifica di equazioni ellittiche del secondo ordine in un dominio limitato aperto con una frattura, su cui è assegnata una condizione al contorno di Dirichlet omogenea. Questo problema locale è correlato a un caso particolare dello studio descritto sopra, in virtù di una forte connessione tra questo tipo di problemi e i problemi al contorno con condizioni miste. Nella presente dissertazione, trattiamo anche una teoria della capillarità non locale. In particolare, consideriamo nuclei di interazione più generali che sono possibilmente anisotropici e non necessariamente invarianti rispetto allo stesso riscalamento. In particolare, la perdita di invarianza è modellata utilizzando due diversi esponenti frazionari per tenere conto della possibilità che il contenitore e l'ambiente presentino caratteristiche diverse rispetto alle interazioni delle particelle. Determiniamo inoltre una legge di Young non locale per l'angolo di contatto tra la gocciolina e la superficie del contenitore e discutiamo la solvibilità e l’unicità della soluzione dell'equazione corrispondente in termini di nuclei di interazione e del relativo coefficiente di adesione.
The aim of the present thesis is to discuss the results obtained during my PhD studies, mainly devoted to nonlocal issues. We first deal with strong unique continuation principles and local asymptotic expansions at certain boundary points for solutions of two different classes of elliptic equations. We start the investigation by a class of fractional elliptic equations in a bounded domain under some outer homogeneous Dirichlet boundary condition. To do this, we exploit the Caffarelli-Silvestre extension procedure, which allows us to get an equivalent formulation of the nonlocal problem as a local problem in one dimension more, consisting in a mixed Dirichlet-Neumann boundary value problem. Then, we use a classical idea by Garofalo and Lin to obtain a doubling-type condition via a monotonicity formula for a suitable Almgren-type frequency function. To overcome the difficulties related to the lack of regularity at the Dirichlet-Neumann junction, we introduce a new technique based on an approximation argument, which leads us to derive a so-called Pohozaev-type identity needed to estimate the derivative of the Almgren function. Thus we gain a strong unique continuation result in the local context, which is in turn combined with blow-up arguments to deduce local asymptotics and, consequently, a strong unique continuation result in the nonlocal setting as well. We also provide a strong unique continuation result from the edge of a crack for the solutions to a specific class of second order elliptic equations in an open bounded domain with a fracture, on which a homogeneous Dirichlet boundary condition is prescribed, in the presence of potentials satisfying either a negligibility condition with respect to the inverse-square weight or some suitable integrability properties. This local problem is related to a particular case of the setting described above, by virtue of a strong connection between this type of problems and the mixed Dirichlet-Neumann boundary value problems. We also treat a capillarity theory of nonlocal type. In our setting, we consider more general interaction kernels that are possibly anisotropic and not necessarily invariant under scaling. In particular, the lack of scale invariance is modeled via two different fractional exponents in order to take into account the possibility that the container and the environment present different features with respect to particle interactions. We determine a nonlocal Young's law for the contact angle between the droplet and the surface of the container and discuss the unique solvability of the corresponding equation in terms of the interaction kernels and of the relative adhesion coefficient.
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Šimkevičiūtė, Jolanta. "Parabolinės lygties su nelokaliąja daugiataške sąlyga sprendimas baigtinių skirtumų metodu." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2011. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2011~D_20110615_110310-18441.

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Darbe nagrinėjamas parabolinių lygčių su nelokaliąja daugiataške sąlyga ir tikrinių reikšmių uždaviniai antrosios eilės paprastajam diferencialiniam operatoriui. Uždavinio specifika yra ta, kad vietoje vienos arba abiejų klasikinių kraštinių sąlygų duota nelokalioji sąlyga. Tokio tipo kraštiniai uždaviniai diferiancialinėms lygtims paskutiniaisiais metais gana intensyviai nagrinėjami matematinėje literatūroje. Darbe naudojamas M. Sapagovo ir A. Štikono 2005 straipsnio metodika tikrinių reikšmių savybėms tirti.
The parabolic equation with nonlocal multi-point condition and the eigenvalue problem for differential operation with nonlocal multi-point condition is investigated in the work. Nonlocal condition is given instead one or both classical boundary conditions. These problems are investigated in the mathematical literature in recent years. The method of analysis to eigenvalue of the article [5] by M. Sapagovas and A. Štikonas on 2005 are used in the work.
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10

Jakubėlienė, Kristina. "Solution of a two-dimensional parabolic equation with an integral condition by the finite-difference method." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2013. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2013~D_20130521_163755-33966.

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The aim of the work is to analyze the finite difference method for solving two-dimensional parabolic equation with an integral boundary condition. The alternating direction method for solving the problem of this kind is analyzed. This method is applied the alternating direction method for solving two-dimensional parabolic equation with two nonlocal integral condition is analyzed. Solution of the problem is found by resolving an additional linear system of equations of lower order . Structure of the spectrum for difference operator with nonlocal condition is analyzed. In order to analyze stability of two-dimensional parabolic equation with one integral condition the structure of spectrum is analyzed. Influence of nonlocal condition for structure of the spectrum is determined. The finite difference method for elliptic problem is constructed.
Darbo tikslas - išnagrinėti dvimatės parabolinio tipo lygties su nelokaliąja integraline sąlyga sprendimą baigtinių skirtumų metodu. Išnagrinėtas kintamųjų krypčių metodo algoritmas tokiam uždaviniui spręsti. Išnagrinėtas dvimatės parabolinės lygties su keliomis nelokaliosiomis integralinėmis kraštinėmis sąlygomis sprendimas kintamųjų krypčių metodu. Uždavinio sprendinys randamas papildomai išsprendžiant neaukštos eilės algebrinę tiesinių lygčių sistemą, kuri sudaroma panaudojant nelokaliąsias integralines sąlygas. Išanalizuota skirtuminio operatoriaus su nelokaliosiomis sąlygomis spektro struktūra. Spektro struktūra išanalizuota tuo tikslu, kad galima būtų išnagrinėti dvimačio parabolinio uždavinio su viena nelokaliąja integraline sąlyga sprendžiamo kintamųjų krypčių ar lokaliai vienmačiu metodu, stabilumą. Nustatyta nelokaliosios sąlygos įtaka spektro struktūrai. Sudarytas elipsinio uždavinio su papildoma nelokaliąja sąlyga sprendimo algoritmas.
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Books on the topic "Nonlocal equation"

1

Shishmarev, I. A. (Ilʹi͡a︡ Andreevich)., ed. Nonlinear nonlocal equations in the theory of waves. Providence, R.I: American Mathematical Society, 1994.

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2

E, Zorumski W., Watson Willie R, and Langley Research Center, eds. Solution of the three-dimensional Helmholtz equation with nonlocal boundary conditions. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1995.

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E, Zorumski W., Watson Willie R, and Langley Research Center, eds. Solution of the three-dimensional Helmholtz equation with nonlocal boundary conditions. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1995.

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Nonlocal quantum field theory and stochastic quantum mechanics. Dordrecht: D. Reidel pub. Co., 1986.

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Nonlocal diffusion problems. Providence, R.I: American Mathematical Society, 2010.

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Andreu-Vaillo, Fuensanta. Nonlocal diffusion problems. Providence, R.I: American Mathematical Society, 2010.

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Nonlocal and abstract parabolic equations and their applications. Warszawa: Institute of Mathematics, Polish Academy of Sciences, 2009.

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Naumkin, P. I. Nonlinear nonlocal equations in the theory of waves. Providence, R.I: American Mathematical Society, 1994.

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E, Zorumski William, and Langley Research Center, eds. Periodic time-domain nonlocal nonreflecting boundary conditions for duct acoustics. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1996.

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E, Zorumski William, and Langley Research Center, eds. Periodic time-domain nonlocal nonreflecting boundary conditions for duct acoustics. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1996.

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Book chapters on the topic "Nonlocal equation"

1

Elmer, F. J., and T. Christen. "The Nonlocal Amplitude Equation." In Partially Intergrable Evolution Equations in Physics, 579–80. Dordrecht: Springer Netherlands, 1990. http://dx.doi.org/10.1007/978-94-009-0591-7_26.

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Chen, Jingkai. "Peridynamics Beam Equation." In Nonlocal Euler–Bernoulli Beam Theories, 9–21. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-69788-4_3.

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Ettoussi, R., Said Melliani, and S. Chadli. "Nonlocal Intuitionistic Fuzzy Differential Equation." In Intuitionistic and Type-2 Fuzzy Logic Enhancements in Neural and Optimization Algorithms: Theory and Applications, 145–53. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-35445-9_13.

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Constantin, P., C. Foias, B. Nicolaenko, and R. Teman. "Application: A Nonlocal Burgers Equation." In Applied Mathematical Sciences, 82–90. New York, NY: Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3506-4_17.

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Stan, Diana, Félix del Teso, and Juan Luis Vázquez. "Porous Medium Equation with Nonlocal Pressure." In Current Research in Nonlinear Analysis, 277–308. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-89800-1_12.

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Bucur, Claudia, and Enrico Valdinoci. "A Nonlocal Nonlinear Stationary Schrödinger Type Equation." In Lecture Notes of the Unione Matematica Italiana, 127–38. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-28739-3_6.

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Boyd, John P. "Envelope Solitary Waves: Third Order Nonlinear Schroedinger Equation and the Klein-Gordon Equation." In Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics, 325–65. Boston, MA: Springer US, 1998. http://dx.doi.org/10.1007/978-1-4615-5825-5_13.

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Boyd, John P. "Water Waves: Fifth-Order Korteweg-Devries Equation." In Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics, 243–78. Boston, MA: Springer US, 1998. http://dx.doi.org/10.1007/978-1-4615-5825-5_10.

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Ván, Péter. "Weakly Nonlocal Non-Equilibrium Thermodynamics: the Cahn-Hilliard Equation." In Advanced Structured Materials, 745–60. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-72440-9_40.

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Andreu-Vaillo, Fuensanta, José Mazón, Julio Rossi, and J. Julián Toledo-Melero. "The Neumann problem for a nonlocal nonlinear diffusion equation." In Mathematical Surveys and Monographs, 99–122. Providence, Rhode Island: American Mathematical Society, 2010. http://dx.doi.org/10.1090/surv/165/05.

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Conference papers on the topic "Nonlocal equation"

1

Bagland, Véronique, and Mohammed Lemou. "Equilibrium states for the Landau-Fermi-Dirac equation." In Nonlocal Elliptic and Parabolic Problems. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2004. http://dx.doi.org/10.4064/bc66-0-2.

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Beneš, Michal, Danielle Hilhorst, and Rémi Weidenfeld. "Interface dynamics for an anisotropic Allen-Cahn equation." In Nonlocal Elliptic and Parabolic Problems. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2004. http://dx.doi.org/10.4064/bc66-0-3.

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Fijałkowski, Piotr, Bogdan Przeradzki, and Robert Stańczy. "A nonlocal elliptic equation in a bounded domain." In Nonlocal Elliptic and Parabolic Problems. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2004. http://dx.doi.org/10.4064/bc66-0-8.

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Laurençot, Philippe. "Steady states for a fragmentation equation with size diffusion." In Nonlocal Elliptic and Parabolic Problems. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2004. http://dx.doi.org/10.4064/bc66-0-14.

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Wellander, Niklas. "Homogenization of a nonlocal electrostatic equation." In 2011 XXXth URSI General Assembly and Scientific Symposium. IEEE, 2011. http://dx.doi.org/10.1109/ursigass.2011.6050346.

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Wolansky, Gershon. "On the equation ut= Δu + M exp u/∫exp udx in planar domains." In Nonlocal Elliptic and Parabolic Problems. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2004. http://dx.doi.org/10.4064/bc66-0-24.

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STAŃCZY, R. "ON SOME ELLIPTIC EQUATION WITH NONLOCAL NONLINEARITY." In Proceedings of the International Conference on Differential Equations. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702067_0045.

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Ashyralyev, Allaberen, Faruk Özger, Theodore E. Simos, George Psihoyios, Ch Tsitouras, and Zacharias Anastassi. "The Hyperbolic-Elliptic Equation with the Nonlocal Condition." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics. AIP, 2011. http://dx.doi.org/10.1063/1.3636797.

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Ru, P., P. K. Jakobsen, and J. V. Moloney. "Nonlocal Adiabatic Elimination in the Maxwell-Bloch Equation." In Nonlinear Dynamics in Optical Systems. Washington, D.C.: Optica Publishing Group, 1992. http://dx.doi.org/10.1364/nldos.1992.mc6.

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Adiabatic elimination is a standard procedure applied to the Maxwell-Bloch laser equations when one variable or more is slaved to the remaining variables. An important case in point is a laser with an extremely large gain bandwidth satisfying the condition γ⊥ ≫ γ||, k where γ⊥ is the polarization dephasing rate, γ|| the de-energization rate and k the cavity damping constant. For example, color center gain media satisfy this criterion and support hundreds of thousands of longitudinal modes in synchronous pumped mode-locking operation. For simple single mode plane wave models the crude adiabatic elimination step of setting the derivative of the polarization variable to zero can be avoided by using center manifold techniques [1]. In this general class of singular perturbation problem, the idea is to coordinatize the problem using linear stability analysis about some known solution and then to construct an approximation to the center manifold on which the (possibly dynamic) solution remains for all time. This procedure has been successfully applied to the Maxwell-Bloch equations describing a single mode homogenously broadened ring laser [2]. Extension of the procedure to nonlinear partial differential equations is very difficult in general as the resulting center manifold may be an infinite dimensional object. When transverse (or additional longitudinal) degrees of freedom are introduced in the Maxwell-Bloch equations in order to investigate spatial pattern formation (or mode-locking dynamics) we find that a crude adiabatic elimination (henceforth referred to as standard adiabatic elimination SAE) leads to nonphysical high transverse (or longitudinal) spatial wavenumber instabilities [3]. Recent attempts to apply the center manifold technique to the transverse problem have met with mixed success [4]. In fact the high transverse wavenumber instability shows an even stronger divergence than the SAE case for positive sign of the laser-atom detuning. Moreover, the analysis becomes unwieldy even in situations when the center manifold approach appears to work.
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Ahmadian, M. T., Abdolreza Pasharavesh, and Ali Fallah. "Application of Nonlocal Theory in Dynamic Pull-In Analysis of Electrostatically Actuated Micro and Nano Beams." In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-48862.

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One of the most important phenomena related to electrically actuated micro and nano electromechanical systems (MEMS\NEMS) is dynamic pull-in instability which occurs when the electrical attraction and beam inertia forces are more than elastic restoring force of the beam. According to failure of classical mechanics constitutive equations in prediction of dynamic behavior of small size systems, nonlocal theory is implemented here to analyze the dynamic pull-in behavior. Equation of motion of an electrostatically actuated micro to nano scale doubly clamped beam is rewritten using differential form of nonlocal theory constitutive equation. To analyze the nonlocal effect equation of motion is nondimentionalized. Governing partial differential equation is transformed to an ordinary differential equation using the Galerkin decomposition method and then is solved implementing differential quadrature method (DQM). Change of dynamic pull-in voltage with respect to size change is investigated. Results indicate as the beam length decreases dynamic pull-in voltage increases due to nonlocal effect and the difference with clasical mechanics results is up to 20% for nano beams.
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Reports on the topic "Nonlocal equation"

1

D'Elia, Marta, Eugenio Aulisa, Giacomo Capodaglio, and Andrea Chierici. Efficient quadrature rules for finite element discretizations of nonlocal equations. Office of Scientific and Technical Information (OSTI), January 2021. http://dx.doi.org/10.2172/1763180.

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Igor D. Kaganovich and Oleg Polomarov. Self-Consistent System of Equations for a Kinetic Description of the Low-Pressure Discharges Accounting for the Nonlocal and Collisionless Electron Dynamics. Office of Scientific and Technical Information (OSTI), May 2003. http://dx.doi.org/10.2172/814016.

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