In this paper, the time-dependent advection-diffusion equation is studied. After introducing these equations in various engineering fields such as gas adsorption, solid dissolution, heat and mass transfer in falling film or pipe and other equations similar to transport phenomena, a new method has been proposed to find their solutions. Among the various works on solving these PDEs by numerical and somewhat analytical methods, a general analytical framework for solving these equations is presented. Using advanced components of Sobolev spaces, weak solutions and some important integral inequalities, an analytical method for the existence and uniqueness of the weak solution of these PDEs is presented, which is the best solution in the proposed structure. Then, with a reduced system of ODE, one can solve the problem of the general parabolic boundary value problem, which includes PDE transport phenomena. Besides, the new approach supports the infinite propagation speed of disturbances of (time-dependent) diffusion-time equations in semi-infinite media.
{"title":"Solving Advection-Diffusion Equations via Sobolev Space Notions","authors":"A. Hasan-Zadeh","doi":"10.12691/IJPDEA-8-1-1","DOIUrl":"https://doi.org/10.12691/IJPDEA-8-1-1","url":null,"abstract":"In this paper, the time-dependent advection-diffusion equation is studied. After introducing these equations in various engineering fields such as gas adsorption, solid dissolution, heat and mass transfer in falling film or pipe and other equations similar to transport phenomena, a new method has been proposed to find their solutions. Among the various works on solving these PDEs by numerical and somewhat analytical methods, a general analytical framework for solving these equations is presented. Using advanced components of Sobolev spaces, weak solutions and some important integral inequalities, an analytical method for the existence and uniqueness of the weak solution of these PDEs is presented, which is the best solution in the proposed structure. Then, with a reduced system of ODE, one can solve the problem of the general parabolic boundary value problem, which includes PDE transport phenomena. Besides, the new approach supports the infinite propagation speed of disturbances of (time-dependent) diffusion-time equations in semi-infinite media.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"59 1","pages":"1-5"},"PeriodicalIF":0.0,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78681182","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 address questions on the existence and multiplicity of solutions to the nonlinear elliptic system in divergence form ⎧⎨ ⎩ div (H∇u) = Hs|∇u|u+[cof∇u]∇P in Ω, det∇u = 1 in Ω, u = φ on ∂Ω. Here H = H(r,s) > 0 is a weight function of class C 2 with Hs = ∂H/∂ s and (r,s) = (|x|, |u|2) , Ω ⊂ Rn is a bounded domain, P = P(x) is an unknown hydrostatic pressure field and φ is a prescribed boundary map. The system is the Euler-Lagrange equation for a weighted Dirichlet energy subject to a pointwise incompressibility constraint on the admissible maps and arises in diverse fields such as geometric function theory and nonlinear elasticity. Whilst the usual methods of critical point theory drastically fail in this vectorial gradient constrained setting we establish the existence of multiple solutions in certain geometries by way of analysing an associated reduced energy for SO(n) -valued fields, a resulting decoupled PDE system and a structure theorem for irrotational vector fields generated by skew-symmetric matrices. Most notably a crucial ”H -condition” linking to the system and precisely capturing an extreme dimensional dichotomy in the structure of the solution set is discovered and analysed. Mathematics subject classification (2010): 35J57, 35J50, 35J62, 49J10, 35A15, 58D19, 22E30.
{"title":"On the existence and multiplicity of topologically twisting incompressible $H$-harmonic maps and a structural H-condition","authors":"George Morrison, A. Taheri","doi":"10.7153/dea-2020-12-04","DOIUrl":"https://doi.org/10.7153/dea-2020-12-04","url":null,"abstract":"In this paper we address questions on the existence and multiplicity of solutions to the nonlinear elliptic system in divergence form ⎧⎨ ⎩ div (H∇u) = Hs|∇u|u+[cof∇u]∇P in Ω, det∇u = 1 in Ω, u = φ on ∂Ω. Here H = H(r,s) > 0 is a weight function of class C 2 with Hs = ∂H/∂ s and (r,s) = (|x|, |u|2) , Ω ⊂ Rn is a bounded domain, P = P(x) is an unknown hydrostatic pressure field and φ is a prescribed boundary map. The system is the Euler-Lagrange equation for a weighted Dirichlet energy subject to a pointwise incompressibility constraint on the admissible maps and arises in diverse fields such as geometric function theory and nonlinear elasticity. Whilst the usual methods of critical point theory drastically fail in this vectorial gradient constrained setting we establish the existence of multiple solutions in certain geometries by way of analysing an associated reduced energy for SO(n) -valued fields, a resulting decoupled PDE system and a structure theorem for irrotational vector fields generated by skew-symmetric matrices. Most notably a crucial ”H -condition” linking to the system and precisely capturing an extreme dimensional dichotomy in the structure of the solution set is discovered and analysed. Mathematics subject classification (2010): 35J57, 35J50, 35J62, 49J10, 35A15, 58D19, 22E30.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"6 1","pages":"47-67"},"PeriodicalIF":0.0,"publicationDate":"2020-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80636818","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}
We present a mathematical analysis of a reaction-diffusion model in a bounded open domain which describes vascular endothelial growth factor(VEGF), endothelial cells and oxygen. We use the parabolic theory to prove the existence of the solution in the function space under the homogeneous Neumann conditions. Then we get the existence of nonnegative solution in by using the global Schauder estimation.
{"title":"Existence and Uniqueness of a Chemotaxis System Influenced by Cancer Cells","authors":"Gang Li, Huijuan Hu, Xi Chen, Feijun Jiang","doi":"10.12691/IJPDEA-7-1-1","DOIUrl":"https://doi.org/10.12691/IJPDEA-7-1-1","url":null,"abstract":"We present a mathematical analysis of a reaction-diffusion model in a bounded open domain which describes vascular endothelial growth factor(VEGF), endothelial cells and oxygen. We use the parabolic theory to prove the existence of the solution in the function space under the homogeneous Neumann conditions. Then we get the existence of nonnegative solution in by using the global Schauder estimation.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"16 1","pages":"1-7"},"PeriodicalIF":0.0,"publicationDate":"2020-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81317264","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 aim of this work is to develop a fuller theory regarding the existence, uniqueness and approximation of solutions to third-order boundary value problems via fixed point methods. To develop this deeper understanding of qualitative properties of solutions, our strategy involves an analysis of the problem under consideration, and its associated operator equations, within closed and bounded sets. This enables our new results to apply to a wider range of problems than those covered in the recent literature and we discuss several examples to illustrate the nature of these advancements. Mathematics subject classification (2010): 34B15.
{"title":"Existence and uniqueness of solutions to third-order boundary value problems: analysis in closed and bounded sets","authors":"S. S. Almuthaybiri, C. Tisdell","doi":"10.7153/dea-2020-12-19","DOIUrl":"https://doi.org/10.7153/dea-2020-12-19","url":null,"abstract":"The aim of this work is to develop a fuller theory regarding the existence, uniqueness and approximation of solutions to third-order boundary value problems via fixed point methods. To develop this deeper understanding of qualitative properties of solutions, our strategy involves an analysis of the problem under consideration, and its associated operator equations, within closed and bounded sets. This enables our new results to apply to a wider range of problems than those covered in the recent literature and we discuss several examples to illustrate the nature of these advancements. Mathematics subject classification (2010): 34B15.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84173545","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, a nonlinear parabolic variational inequality in noncylindrical domain is considered. Using extended Rothe’s method recently achieved in [11] an approximate solution is constructed. Existence and uniqueness results are proved. Also, we present some further results and comments related to the main result. Mathematics subject classification (2010): 65N40, 35K20.
{"title":"Rothe's method for nonlinear parabolic variational inequalities in noncylindrical domains","authors":"G. Kulieva, K. Kuliev","doi":"10.7153/dea-2020-12-15","DOIUrl":"https://doi.org/10.7153/dea-2020-12-15","url":null,"abstract":"In this paper, a nonlinear parabolic variational inequality in noncylindrical domain is considered. Using extended Rothe’s method recently achieved in [11] an approximate solution is constructed. Existence and uniqueness results are proved. Also, we present some further results and comments related to the main result. Mathematics subject classification (2010): 65N40, 35K20.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"40 1","pages":"227-242"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88358752","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}
Using two fractional-order integral inequalities we investigate the existence and uniqueness of solutions of the fractional nonlinear Volterra integral equation and the fractional nonlinear integrodifferential equation in Banach space Cξ , using an adequate norm, || · ||ξ ,∞ . Besides, we study the solution estimate and investigate their continuous dependence. Mathematics subject classification (2010): 26A33, 34A08, 34A12, 34A60, 34G20.
{"title":"Existence results and continuity dependence of solutions for fractional equations","authors":"J. V. C. Sousa","doi":"10.7153/dea-2020-12-24","DOIUrl":"https://doi.org/10.7153/dea-2020-12-24","url":null,"abstract":"Using two fractional-order integral inequalities we investigate the existence and uniqueness of solutions of the fractional nonlinear Volterra integral equation and the fractional nonlinear integrodifferential equation in Banach space Cξ , using an adequate norm, || · ||ξ ,∞ . Besides, we study the solution estimate and investigate their continuous dependence. Mathematics subject classification (2010): 26A33, 34A08, 34A12, 34A60, 34G20.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"30 1","pages":"377-396"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83357454","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 work, an illustrative discussion have been made on unbounded oscillation properties of a class of fourth order neutral functional difference equations of the form: Δ2(r(n)Δ2(y(n)+ p(n)y(n− τ)))+g(n)G(y(n−σ))−h(n)H(y(n−α)) = 0 under the assumptions ∞ ∑ n=0 n r(n) = ∞, ∞ ∑ n=0 n r(n) < ∞. New oscillation criteria have been established for different ranges of p(n) with |p(n)| < ∞ . Mathematics subject classification (2010): 39A10, 39A12.
本文在假设∞∑n=0 n r(n) =∞，∞∑n=0 n r(n) =∞，∞∑n=0 n r(n) <∞的条件下，讨论了形式为Δ2(r(n)Δ2(y(n)+ p(n)y(n−τ)) +g(n) g(y(n−σ)) - h(n) h(y(n−α)) =0的四阶中立型泛函差分方程的无界振荡性质。在p(n)| <∞的情况下，对p(n)的不同范围建立了新的振荡判据。数学学科分类(2010):39A10, 39A12。
{"title":"On unbounded oscillation of fourth order functional difference equations","authors":"A. Tripathy","doi":"10.7153/dea-2020-12-17","DOIUrl":"https://doi.org/10.7153/dea-2020-12-17","url":null,"abstract":"In this work, an illustrative discussion have been made on unbounded oscillation properties of a class of fourth order neutral functional difference equations of the form: Δ2(r(n)Δ2(y(n)+ p(n)y(n− τ)))+g(n)G(y(n−σ))−h(n)H(y(n−α)) = 0 under the assumptions ∞ ∑ n=0 n r(n) = ∞, ∞ ∑ n=0 n r(n) < ∞. New oscillation criteria have been established for different ranges of p(n) with |p(n)| < ∞ . Mathematics subject classification (2010): 39A10, 39A12.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"19 1","pages":"259-275"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76340479","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 aim of this paper is to study a stage-structured pest management model with mixed type of functional response i.e., Holling type-I and Beddington-DeAngelis functional response with impulsive biological control. Stage structuring is proposed due to the fact that almost all the pests in their life pass through two stages namely, immature larva and mature adult. It is assumed that immature susceptible pests and exposed pests are attacked by a natural enemy and susceptible pests (immature and mature) are contacted by infected pests which make them exposed. Infected pests and natural enemies are infused impulsively after fixed intervals. All positive solutions are proved to be uniformly ultimately bounded. The stability analysis of pest extinction periodic solution, as well as the permanence of system, are obtained by making use of floquet’s theory, small amplitude perturbation technique, and comparison theorem. The results obtained provide certain dependable theoretical findings for effective pest management. At last, theoretical findings are confirmed by means of numerical simulation. Mathematics subject classification (2010): 92D25, 34C11.
{"title":"Analysis of stage-structured model with mixed type of functional response and impulsive biological control","authors":"Bhanu Gupta, Amit Sharma, J. Dhar, S. Srivastava","doi":"10.7153/DEA-2020-12-05","DOIUrl":"https://doi.org/10.7153/DEA-2020-12-05","url":null,"abstract":"The aim of this paper is to study a stage-structured pest management model with mixed type of functional response i.e., Holling type-I and Beddington-DeAngelis functional response with impulsive biological control. Stage structuring is proposed due to the fact that almost all the pests in their life pass through two stages namely, immature larva and mature adult. It is assumed that immature susceptible pests and exposed pests are attacked by a natural enemy and susceptible pests (immature and mature) are contacted by infected pests which make them exposed. Infected pests and natural enemies are infused impulsively after fixed intervals. All positive solutions are proved to be uniformly ultimately bounded. The stability analysis of pest extinction periodic solution, as well as the permanence of system, are obtained by making use of floquet’s theory, small amplitude perturbation technique, and comparison theorem. The results obtained provide certain dependable theoretical findings for effective pest management. At last, theoretical findings are confirmed by means of numerical simulation. Mathematics subject classification (2010): 92D25, 34C11.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"21 1","pages":"69-88"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91084707","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}
{"title":"Multiplicity of solutions for a fractional p-Kirchhoff type problem with sign-changing weights function","authors":"Yu an Gui","doi":"10.7153/dea-2020-12-09","DOIUrl":"https://doi.org/10.7153/dea-2020-12-09","url":null,"abstract":"","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"50 1","pages":"129-142"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73819093","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 study, fractional reduced differential transform method (FRDTM) is developed to derive a semianalytical solution of fractional partial differential equations which involves Riesz space fractional derivatives. We focus primarily on implementing the novel algorithm of FRDTM to Riesz space -fractional telegraph equation while the telegraph equation has fractional order. Some theorems with their proofs which are used for calculating differential transform of Riesz derivative operator are presented, as well as the convergence condition and the error bound of the proposed method are established. To illustrate the reliability and capability of the method, some examples are provided. The results reveal that the algorithm is very effective and uncomplicated. Mathematics subject classification (2010): 65Z05, 35Q60, 35Q99.
{"title":"Analytical approximation of time-fractional telegraph equation with Riesz space-fractional derivative","authors":"S. Mohammadian, Y. Mahmoudi, F. D. Saei","doi":"10.7153/dea-2020-12-16","DOIUrl":"https://doi.org/10.7153/dea-2020-12-16","url":null,"abstract":"In this study, fractional reduced differential transform method (FRDTM) is developed to derive a semianalytical solution of fractional partial differential equations which involves Riesz space fractional derivatives. We focus primarily on implementing the novel algorithm of FRDTM to Riesz space -fractional telegraph equation while the telegraph equation has fractional order. Some theorems with their proofs which are used for calculating differential transform of Riesz derivative operator are presented, as well as the convergence condition and the error bound of the proposed method are established. To illustrate the reliability and capability of the method, some examples are provided. The results reveal that the algorithm is very effective and uncomplicated. Mathematics subject classification (2010): 65Z05, 35Q60, 35Q99.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"16 1","pages":"243-258"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89431770","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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