The study of coupled systems of hybrid fractional differential equations requires the attention of scientists for the exploration of their different important aspects. Our aim in this paper is to study the existence and uniqueness of the solution for impulsive hybrid fractional differential equations. The novelty of this work is the study of a coupled system of impulsive hybrid fractional differential equations with initial and boundary hybrid conditions. We used the classical fixed-point theorems such as the Banach fixed-point theorem and Leray–Schauder alternative fixed-point theorem for existence results. We also give an example of the main results.
{"title":"Investigation of a Mild Solution to Coupled Systems of Impulsive Hybrid Fractional Differential Equations","authors":"M. Hannabou, Hilal Khalid","doi":"10.1155/2019/2618982","DOIUrl":"https://doi.org/10.1155/2019/2618982","url":null,"abstract":"The study of coupled systems of hybrid fractional differential equations requires the attention of scientists for the exploration of their different important aspects. Our aim in this paper is to study the existence and uniqueness of the solution for impulsive hybrid fractional differential equations. The novelty of this work is the study of a coupled system of impulsive hybrid fractional differential equations with initial and boundary hybrid conditions. We used the classical fixed-point theorems such as the Banach fixed-point theorem and Leray–Schauder alternative fixed-point theorem for existence results. We also give an example of the main results.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2019/2618982","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49429055","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}
Wondwosen Gebeyaw Melesse, A. Tiruneh, G. A. Derese
In this paper, an initial value method for solving a class of linear second-order singularly perturbed differential difference equation containing mixed shifts is proposed. In doing so, first, the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay and advance parameters using Taylor series expansion. From the modified problem, two explicit initial value problems which are independent of the perturbation parameter are produced; namely, the reduced problem and the boundary layer correction problem. These problems are then solved analytically and/or numerically, and those solutions are combined to give an approximate solution to the original problem. An error estimate for this method is derived using maximum norm. Several test problems are considered to illustrate the theoretical results. It is observed that the present method approximates the exact solution very well.
{"title":"Solving Linear Second-Order Singularly Perturbed Differential Difference Equations via Initial Value Method","authors":"Wondwosen Gebeyaw Melesse, A. Tiruneh, G. A. Derese","doi":"10.1155/2019/5259130","DOIUrl":"https://doi.org/10.1155/2019/5259130","url":null,"abstract":"In this paper, an initial value method for solving a class of linear second-order singularly perturbed differential difference equation containing mixed shifts is proposed. In doing so, first, the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay and advance parameters using Taylor series expansion. From the modified problem, two explicit initial value problems which are independent of the perturbation parameter are produced; namely, the reduced problem and the boundary layer correction problem. These problems are then solved analytically and/or numerically, and those solutions are combined to give an approximate solution to the original problem. An error estimate for this method is derived using maximum norm. Several test problems are considered to illustrate the theoretical results. It is observed that the present method approximates the exact solution very well.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":"2019 1","pages":"1-10"},"PeriodicalIF":1.6,"publicationDate":"2019-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2019/5259130","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45269739","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}
Our purpose in this paper is to prove, under some regularity conditions on the data, the solvability in a Gevrey class of bound −1 on the interval −1,1 of a class of nonlinear fractional functional differential equations.
{"title":"Solvability in Gevrey Classes of Some Nonlinear Fractional Functional Differential Equations","authors":"Hicham Zoubeir","doi":"10.1155/2020/3739249","DOIUrl":"https://doi.org/10.1155/2020/3739249","url":null,"abstract":"Our purpose in this paper is to prove, under some regularity conditions on the data, the solvability in a Gevrey class of bound −1 on the interval −1,1 of a class of nonlinear fractional functional differential equations.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2020/3739249","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46461069","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}
This paper formulates sufficiency-type global stability and asymptotic stability results for, in general, nonlinear time-varying dynamic systems with state-trajectory solution-dependent parameterizations. The stability proofs are based on obtaining sufficiency-type conditions which guarantee that either the norms of the solution trajectory or alternative interval-type integrals of the matrix of dynamics of the higher-order than linear terms do not grow faster than their available supremum on the preceding time intervals. Some extensions are also given based on the use of a truncated Taylor series expansion of chosen truncation order with multiargument integral remainder for the dynamics of the differential system.
{"title":"On Some Sufficiency-Type Global Stability Results for Time-Varying Dynamic Systems with State-Dependent Parameterizations","authors":"M. de La Sen","doi":"10.1155/2019/5097974","DOIUrl":"https://doi.org/10.1155/2019/5097974","url":null,"abstract":"This paper formulates sufficiency-type global stability and asymptotic stability results for, in general, nonlinear time-varying dynamic systems with state-trajectory solution-dependent parameterizations. The stability proofs are based on obtaining sufficiency-type conditions which guarantee that either the norms of the solution trajectory or alternative interval-type integrals of the matrix of dynamics of the higher-order than linear terms do not grow faster than their available supremum on the preceding time intervals. Some extensions are also given based on the use of a truncated Taylor series expansion of chosen truncation order with multiargument integral remainder for the dynamics of the differential system.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2019/5097974","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46903943","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}
Saleh Alshammari, M. Al‐Smadi, Mohammad Al Shammari, I. Hashim, M. A. Alias
In this article, an analytical reliable treatment based on the concept of residual error functions is employed to address the series solution of the differential logistic system in the fractional sense. The proposed technique is a combination of the generalized Taylor series and minimizing the residual error function. The solution methodology depends on the generation of a fractional expansion in an effective convergence formula, as well as on the optimization of truncated errors,Resqjt, through the use of repeated Caputo derivatives without any restrictive assumptions of system nature. To achieve this, some logistic patterns are tested to demonstrate the reliability and applicability of the suggested approach. Numerical comparison depicts that the proposed technique has high accuracy and less computational effect and is more efficient.
{"title":"Advanced Analytical Treatment of Fractional Logistic Equations Based on Residual Error Functions","authors":"Saleh Alshammari, M. Al‐Smadi, Mohammad Al Shammari, I. Hashim, M. A. Alias","doi":"10.1155/2019/7609879","DOIUrl":"https://doi.org/10.1155/2019/7609879","url":null,"abstract":"In this article, an analytical reliable treatment based on the concept of residual error functions is employed to address the series solution of the differential logistic system in the fractional sense. The proposed technique is a combination of the generalized Taylor series and minimizing the residual error function. The solution methodology depends on the generation of a fractional expansion in an effective convergence formula, as well as on the optimization of truncated errors,Resqjt, through the use of repeated Caputo derivatives without any restrictive assumptions of system nature. To achieve this, some logistic patterns are tested to demonstrate the reliability and applicability of the suggested approach. Numerical comparison depicts that the proposed technique has high accuracy and less computational effect and is more efficient.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2019/7609879","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44465391","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 mathematical model of a contact problem in thermo-electro-viscoelasticity with the normal compliance conditions and Tresca’s friction law. We present a variational formulation of the problem, and we prove the existence and uniqueness of the weak solution. We also study the numerical approach using spatially semidiscrete and fully discrete finite element schemes with Euler’s backward scheme. Finally, we derive error estimates on the approximate solutions.
{"title":"Variational and Numerical Analysis for Frictional Contact Problem with Normal Compliance in Thermo-Electro-Viscoelasticity","authors":"M. Bouallala, E. Essoufi, M. Alaoui","doi":"10.1155/2019/6972742","DOIUrl":"https://doi.org/10.1155/2019/6972742","url":null,"abstract":"In this paper, we consider a mathematical model of a contact problem in thermo-electro-viscoelasticity with the normal compliance conditions and Tresca’s friction law. We present a variational formulation of the problem, and we prove the existence and uniqueness of the weak solution. We also study the numerical approach using spatially semidiscrete and fully discrete finite element schemes with Euler’s backward scheme. Finally, we derive error estimates on the approximate solutions.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2019/6972742","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43855438","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 expand an SIR epidemic model with vertical and nonlinear incidence rates from a deterministic frame to a stochastic one. The existence of a positive global analytical solution of the proposed stochastic model is shown, and conditions for the extinction and persistence of the disease are established. The presented results are demonstrated by numerical simulations.
{"title":"Analysis of a Stochastic SIR Model with Vaccination and Nonlinear Incidence Rate","authors":"A. El koufi, Jihad Adnani, A. Bennar, N. Yousfi","doi":"10.1155/2019/9275051","DOIUrl":"https://doi.org/10.1155/2019/9275051","url":null,"abstract":"We expand an SIR epidemic model with vertical and nonlinear incidence rates from a deterministic frame to a stochastic one. The existence of a positive global analytical solution of the proposed stochastic model is shown, and conditions for the extinction and persistence of the disease are established. The presented results are demonstrated by numerical simulations.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2019/9275051","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43083584","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 intuitionistic fuzzy partial functional differential equations with local and nonlocal initial conditions using the Banach fixed point theorem. A new complete intuitionistic fuzzy metric space is proposed to investigate the existence and uniqueness of intuitionistic fuzzy solutions for these problems. We use the level-set representation of intuitionistic fuzzy functions and define the solution to an intuitionistic fuzzy partial functional differential equation problem through a corresponding parametric problem and further develop theoretical results on the existence and uniqueness of the solution. An example is presented to illustrate the results with some numerical simulation for α-cuts of the intuitionistic fuzzy solutions: we give the representation of the surface of intuitionistic fuzzy solutions.
{"title":"The Existence and Uniqueness of Intuitionistic Fuzzy Solutions for Intuitionistic Fuzzy Partial Functional Differential Equations","authors":"B. Ben Amma, S. Melliani, L. S. Chadli","doi":"10.1155/2019/9210641","DOIUrl":"https://doi.org/10.1155/2019/9210641","url":null,"abstract":"In this paper, we consider intuitionistic fuzzy partial functional differential equations with local and nonlocal initial conditions using the Banach fixed point theorem. A new complete intuitionistic fuzzy metric space is proposed to investigate the existence and uniqueness of intuitionistic fuzzy solutions for these problems. We use the level-set representation of intuitionistic fuzzy functions and define the solution to an intuitionistic fuzzy partial functional differential equation problem through a corresponding parametric problem and further develop theoretical results on the existence and uniqueness of the solution. An example is presented to illustrate the results with some numerical simulation for α-cuts of the intuitionistic fuzzy solutions: we give the representation of the surface of intuitionistic fuzzy solutions.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2019/9210641","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42930614","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}
This paper is concerned with a delayed model of mutual interactions between the economically active population and the economic growth. The main purpose is to investigate the direction and stability of the bifurcating branch resulting from the increase of delay. By using a second order approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points and we show that the system under consideration can undergo a supercritical or subcritical Hopf bifurcation and the bifurcating periodic solution is stable or unstable in a neighborhood of some bifurcation points, depending on the choice of parameters.
{"title":"Direction and Stability of Hopf Bifurcation in a Delayed Solow Model with Labor Demand","authors":"S. ElFadily, A. Kaddar, K. Najib","doi":"10.1155/2019/7609828","DOIUrl":"https://doi.org/10.1155/2019/7609828","url":null,"abstract":"This paper is concerned with a delayed model of mutual interactions between the economically active population and the economic growth. The main purpose is to investigate the direction and stability of the bifurcating branch resulting from the increase of delay. By using a second order approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points and we show that the system under consideration can undergo a supercritical or subcritical Hopf bifurcation and the bifurcating periodic solution is stable or unstable in a neighborhood of some bifurcation points, depending on the choice of parameters.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2019/7609828","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45597053","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 research, a new approach is presented for solving delay differential equations (DDEs) which is a blend of Sumudu transform and variational iteration method (VIM). A general Lagrange multiplier is used to construct a correction functional. This is done with an uncommon Sumudu transform alongside variational theory. A few numerical cases were solved to demonstrate methodology of this new approach. Objective of this research is to reduce the complexity of computational work compared to the conventional approaches. It can be concluded that the amount of evaluation is reduced but at the same time the results are comparable as in the previous works.
{"title":"Variational Iteration Method and Sumudu Transform for Solving Delay Differential Equation","authors":"S. Vilu, R. Ahmad, U. S. Din","doi":"10.1155/2019/6306120","DOIUrl":"https://doi.org/10.1155/2019/6306120","url":null,"abstract":"In this research, a new approach is presented for solving delay differential equations (DDEs) which is a blend of Sumudu transform and variational iteration method (VIM). A general Lagrange multiplier is used to construct a correction functional. This is done with an uncommon Sumudu transform alongside variational theory. A few numerical cases were solved to demonstrate methodology of this new approach. Objective of this research is to reduce the complexity of computational work compared to the conventional approaches. It can be concluded that the amount of evaluation is reduced but at the same time the results are comparable as in the previous works.","PeriodicalId":55967,"journal":{"name":"International Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1155/2019/6306120","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44963766","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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