In many cases, groundwater flow in an unconfined aquifer can be simplified to a one-dimensional Sturm-Liouville model of the form: begin{equation*} x''(t)+lambda x(t)=h(t)+varepsilon f(x(t)),hspace{.1in}tin(0,pi) end{equation*} subject to non-local boundary conditions begin{equation*} x(0)=h_1+varepsiloneta_1(x)text{ and } x(pi)=h_2+varepsiloneta_2(x). end{equation*} In this paper, we study the existence of solutions to the above Sturm-Liouville problem under the assumption that $varepsilon$ is a small parameter. Our method will be analytical, utilizing the implicit function theorem and its generalizations.
在许多情况下，无承压含水层中的地下水流动可以简化为一维Sturm-Liouville模型，其形式为:begin{equation*} x''(t)+lambda x(t)=h(t)+varepsilon f(x(t)),hspace{.1in}tin(0,pi) end{equation*}在非局部边界条件下begin{equation*} x(0)=h_1+varepsiloneta_1(x)text{ and } x(pi)=h_2+varepsiloneta_2(x). end{equation*}。本文在$varepsilon$为小参数的假设下，研究上述Sturm-Liouville问题解的存在性。我们的方法是解析式的，利用隐函数定理及其推广。
{"title":"Weakly nonlocal boundary value problems with application to geology","authors":"D. Maroncelli, E. Collins","doi":"10.7153/DEA-2021-13-12","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-12","url":null,"abstract":"In many cases, groundwater flow in an unconfined aquifer can be simplified to a one-dimensional Sturm-Liouville model of the form: begin{equation*} x''(t)+lambda x(t)=h(t)+varepsilon f(x(t)),hspace{.1in}tin(0,pi) end{equation*} subject to non-local boundary conditions begin{equation*} x(0)=h_1+varepsiloneta_1(x)text{ and } x(pi)=h_2+varepsiloneta_2(x). end{equation*} In this paper, we study the existence of solutions to the above Sturm-Liouville problem under the assumption that $varepsilon$ is a small parameter. Our method will be analytical, utilizing the implicit function theorem and its generalizations.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48580532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present results which allow us to establish the existence of solutions to nonlinear Sturm-Liouville problems with unbounded nonlinearities. We consider both regular and singular problems. Our main results rely on a variant of the Lyapunov-Schmidt used in conjunction with topological degree theory. Mathematics subject classification (2010): 34A34, 34B15, 47H11.
{"title":"Existence of solutions to nonlinear Sturm-Liouville problems with large nonlinearities","authors":"B. Freedman, Jesús F. Rodríguez","doi":"10.7153/DEA-2021-13-11","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-11","url":null,"abstract":"In this paper, we present results which allow us to establish the existence of solutions to nonlinear Sturm-Liouville problems with unbounded nonlinearities. We consider both regular and singular problems. Our main results rely on a variant of the Lyapunov-Schmidt used in conjunction with topological degree theory. Mathematics subject classification (2010): 34A34, 34B15, 47H11.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider a class of fractional p -Laplacian system with three fractional critical Sobolev-Hardy exponents. By the Ekeland variational principle and the MountainPass theorem, we study the existence and multiplicity of positive solutions to the system. Mathematics subject classification (2010): 35B33, 35J60, 35J65.
{"title":"Solutions for the fractional p-Laplacian systems with several critical Sobolev-Hardy terms","authors":"I. Dehsari, N. Nyamoradi","doi":"10.7153/DEA-2021-13-02","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-02","url":null,"abstract":"In this paper, we consider a class of fractional p -Laplacian system with three fractional critical Sobolev-Hardy exponents. By the Ekeland variational principle and the MountainPass theorem, we study the existence and multiplicity of positive solutions to the system. Mathematics subject classification (2010): 35B33, 35J60, 35J65.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Martinez, E. V. Castelani, C. Martinez, G. Bressan, Roberto Molina de Souza
. In this work, we consider a third order equation of three points with non-homogeneous conditions at the border. We apply Avery Peterson’s theorem, and present a theoretical result that guarantees the existence of multiple solutions to this problem under certain conditions. In addition, we present non-trivial examples and a new numerical method based on optimization is introduced.
{"title":"Multiple solutions to a third-order three-point nonhomogeneous boundary value problem aided by nonlinear programming methods","authors":"A. Martinez, E. V. Castelani, C. Martinez, G. Bressan, Roberto Molina de Souza","doi":"10.7153/DEA-2021-13-03","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-03","url":null,"abstract":". In this work, we consider a third order equation of three points with non-homogeneous conditions at the border. We apply Avery Peterson’s theorem, and present a theoretical result that guarantees the existence of multiple solutions to this problem under certain conditions. In addition, we present non-trivial examples and a new numerical method based on optimization is introduced.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. The interactions among delta shock waves, vacuum states and contact discontinuities for the equations of constant pressure fl uid dynamics are analyzed. By solving the Riemann problem with initial data of three piecewise constant states case by case, the global structures of solutions with four different con fi gurations are constructed. Furthermore, the numerical simula-tions completely coinciding with theoretical analysis are presented.
{"title":"Interactions of delta shock waves for the equations of constant pressure fluid dynamics","authors":"Yu Zhang, Yan Zhang","doi":"10.7153/DEA-2021-13-05","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-05","url":null,"abstract":". The interactions among delta shock waves, vacuum states and contact discontinuities for the equations of constant pressure fl uid dynamics are analyzed. By solving the Riemann problem with initial data of three piecewise constant states case by case, the global structures of solutions with four different con fi gurations are constructed. Furthermore, the numerical simula-tions completely coinciding with theoretical analysis are presented.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130921","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main subject of this paper is the study of third order linear partial differential equations with analytic coefficients in a two variables domain. We aim the existence of solutions by algorithmic means, in the real or complex analytical case. This is done by introducing methods inspired by the classical method of Frobenius method for analytic second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial cubic, which is an affine plane curve of degree three. Points in this curve are associate to solutions of the Euler PDE. Then comes the concept of regular singularity for the PDE, followed by a notion of resonance and a partial classification of PDEs having such regular singularities. Finally, we obtain convergence theorems, which must necessarily take into account the existence of resonances and the type of PDE (parabolic, elliptical or hyperbolic). We provide some examples of PDEs that may be treated with our methods. This is the first study in this rich subject. Our results are a first step in the reintroduction of techniques from ordinary differential equations in the study of classical problems involving partial differential equations. Our solutions are constructive and computationally viable. Mathematics subject classification (2010): 35A20, 35A24, 35A30, 35C10.
{"title":"A complete Frobenius type method for linear partial differential equations of third order","authors":"V. León, B. Scárdua","doi":"10.7153/DEA-2021-13-08","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-08","url":null,"abstract":"The main subject of this paper is the study of third order linear partial differential equations with analytic coefficients in a two variables domain. We aim the existence of solutions by algorithmic means, in the real or complex analytical case. This is done by introducing methods inspired by the classical method of Frobenius method for analytic second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial cubic, which is an affine plane curve of degree three. Points in this curve are associate to solutions of the Euler PDE. Then comes the concept of regular singularity for the PDE, followed by a notion of resonance and a partial classification of PDEs having such regular singularities. Finally, we obtain convergence theorems, which must necessarily take into account the existence of resonances and the type of PDE (parabolic, elliptical or hyperbolic). We provide some examples of PDEs that may be treated with our methods. This is the first study in this rich subject. Our results are a first step in the reintroduction of techniques from ordinary differential equations in the study of classical problems involving partial differential equations. Our solutions are constructive and computationally viable. Mathematics subject classification (2010): 35A20, 35A24, 35A30, 35C10.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"99 6 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper, we deal with the existence and uniqueness of solutions of a coupled system of nonlinear implicit fractional differential equations of Caputo-type modi fi cation of the Erd´elyi-Kober involving both retarded and advanced arguments. The arguments are based upon the Banach contraction principle and Schauder’s fi xed point theorem. An example is included to show the applicability of our outcomes.
{"title":"Caputo type modification of the Erdélyi-Kober coupled implicit fractional differential systems with retardation and anticipation","authors":"Mokhtar Boumaaza, M. Benchohra, J. Nieto","doi":"10.7153/DEA-2021-13-07","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-07","url":null,"abstract":". In this paper, we deal with the existence and uniqueness of solutions of a coupled system of nonlinear implicit fractional differential equations of Caputo-type modi fi cation of the Erd´elyi-Kober involving both retarded and advanced arguments. The arguments are based upon the Banach contraction principle and Schauder’s fi xed point theorem. An example is included to show the applicability of our outcomes.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Prof. Dr. Mohamed Abdalla Darwish, M. Metwali, D. Regan
. In this paper we study the existence of monotonic solutions of fractional nonlinear quadratic integral equations in the space of Lebesgue integrable functions on [ 0 , τ ] . The unique-ness of the solution is also discussed. In addition an example is given to illustrate our abstract results.
{"title":"Unique solvability of fractional quadratic nonlinear integral equations","authors":"Prof. Dr. Mohamed Abdalla Darwish, M. Metwali, D. Regan","doi":"10.7153/DEA-2021-13-01","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-01","url":null,"abstract":". In this paper we study the existence of monotonic solutions of fractional nonlinear quadratic integral equations in the space of Lebesgue integrable functions on [ 0 , τ ] . The unique-ness of the solution is also discussed. In addition an example is given to illustrate our abstract results.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Existence and uniqueness of strong solutions for three dimensional system of globally modi fi ed magnetohydrodynamics equations containing in fi nite delays terms are established together with some qualitative properties of the solution in this work. The existence is proved by making use of Galerkin’s method, Cauchy-Lipshitz’s theorem, a priori estimates, the Aubin-Lions compactness theorem. Moreover, we study the asymptotic behavior of the solution.
{"title":"Three dimensional system of globally modified magnetohydrodynamics equations with infinite delays","authors":"G. Deugoue, J. K. Djoko, A. C. Fouape","doi":"10.7153/dea-2021-13-21","DOIUrl":"https://doi.org/10.7153/dea-2021-13-21","url":null,"abstract":". Existence and uniqueness of strong solutions for three dimensional system of globally modi fi ed magnetohydrodynamics equations containing in fi nite delays terms are established together with some qualitative properties of the solution in this work. The existence is proved by making use of Galerkin’s method, Cauchy-Lipshitz’s theorem, a priori estimates, the Aubin-Lions compactness theorem. Moreover, we study the asymptotic behavior of the solution.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sufficient conditions for the existence of solutions to a coupled system of fractionalorder differential inclusions associated with fractional non-separated boundary conditions for multivalued maps are established, by employing the nonlinear alternative of Leray–Schauder type. We emphasize that the methods of fixed point theory used in our analysis are standard, although their application to a system of fractional-order differential inclusions is new. Mathematics subject classification (2010): 34A08, 34A60, 34B15.
{"title":"Existence of solutions for a coupled system of Caputo type fractional-order differential inclusions with non-separated boundary conditions on multivalued maps","authors":"B. Krushna, K. R. Prasad, P. Veeraiah","doi":"10.7153/DEA-2021-13-10","DOIUrl":"https://doi.org/10.7153/DEA-2021-13-10","url":null,"abstract":"Sufficient conditions for the existence of solutions to a coupled system of fractionalorder differential inclusions associated with fractional non-separated boundary conditions for multivalued maps are established, by employing the nonlinear alternative of Leray–Schauder type. We emphasize that the methods of fixed point theory used in our analysis are standard, although their application to a system of fractional-order differential inclusions is new. Mathematics subject classification (2010): 34A08, 34A60, 34B15.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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