The problems of stability system that arises in the construction of different automatic systems of indirect control are considered. It is known that a given program is not always exactly performed, since there are always initial, constantly acting perturbations. Therefore, it is also reasonable to require the stability of the program manifold itself with respect to some function. In the first part, the stability being investigated of automatic indirect control systems with rigid and tachometric feedback. Necessary and sufficient conditions for the absolute stability of a program manifold are established separately. In the second part, the automatic systems of indirect control taking into account the external load are considered. The equations of the hydraulic actuator, taking into account the action of an external load, are presented in a convenient form for research. Then it reduces to studying the stability of the system of equations with respect to a given program manifold. By constructing LyapunovВ’s functions for the system in canonical form, sufficient conditions are obtained for the absolute stability of the program manifold. The results obtained can be used in the construction of stable automatic indirect control systems.
{"title":"Stability of the program manifold of automatic indirect control systems taking into account the external load","authors":"S. Zhumatov, S. Mynbayeva","doi":"10.31197/atnaa.1200890","DOIUrl":"https://doi.org/10.31197/atnaa.1200890","url":null,"abstract":"The problems of stability system that arises in the construction of different automatic systems of indirect control are considered. It is known that a given program is not always exactly performed, since there are always initial, constantly acting perturbations. \u0000Therefore, it is also reasonable to require the stability of the program manifold itself with respect to some function. In the first part, the stability being investigated of automatic indirect control systems with rigid and tachometric feedback. Necessary and sufficient conditions for the absolute stability of a program manifold are established separately. In the second part, the automatic systems of indirect control taking into account the external load are considered. The equations of the hydraulic actuator, taking into account the action of an external load, are presented in a convenient form for research. Then it reduces to studying the stability of the system of equations with respect to a given program manifold. By constructing LyapunovВ’s functions for the system in canonical form, sufficient conditions are obtained for the absolute stability of the program manifold. The results obtained can be used in the construction of stable automatic indirect control systems.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85882420","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 initiate the study of fixed points for interpolative mappings in $m$-metric spaces. We discuss three different cases: the sum of textquotedblleft interpolative exponents" is less than, equal to or greater than 1. We support each of our result by examples in $m$% -metric spaces. In the last section, we obtain our results in $p$-metric spaces. Finally we note that our results generalize results of cite{EY}, cite{GH} and cite{K} from ordinary metric to $m$- and $p$-metrics.
本文研究了$m$ -度量空间内插映射的不动点问题。我们讨论了三种不同的情况:textquotedblleft插值指数”的和小于、等于或大于1。中的例子支持了我们的每一个结果 $m$% -metric spaces. In the last section, we obtain our results in $p$-metric spaces. Finally we note that our results generalize results of cite{EY}, cite{GH} and cite{K} from ordinary metric to $m$- and $p$-metrics.
{"title":"Interpolative Contractive Results for $m$-Metric Spaces","authors":"","doi":"10.31197/atnaa.1220114","DOIUrl":"https://doi.org/10.31197/atnaa.1220114","url":null,"abstract":"In this paper, we initiate the study of fixed points for \u0000interpolative mappings in $m$-metric spaces. We discuss three different \u0000cases: the sum of textquotedblleft interpolative exponents\" is less than, \u0000equal to or greater than 1. We support each of our result by examples in $m$% \u0000-metric spaces. In the last section, we obtain our results in $p$-metric \u0000spaces. Finally we note that \u0000our results generalize results of cite{EY}, cite{GH} and cite{K} from ordinary metric to $m$- and $p$-metrics.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73013123","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}
Bell's polynomials have been used in many different fields, ranging from number theory to operators theory. In this article we show a method to compute the Laplace Transform (LT) of nested analytic functions. To this aim, we provide a table of the first few values of the complete Bell's polynomials, which are then used to evaluate the LT of composite exponential functions. Furthermore a code for approximating the Laplace Transform of general analytic composite functions is created and presented. A graphical verification of the proposed technique is illustrated in the last section.
{"title":"Laplace Transform of nested analytic functions via Bell’s polynomials","authors":"P. Ricci, D. Caratelli, S. Pinelas","doi":"10.31197/atnaa.1187617","DOIUrl":"https://doi.org/10.31197/atnaa.1187617","url":null,"abstract":"Bell's polynomials have been used in many different fields, ranging from number theory to operators theory. In this article we show a method to compute the Laplace Transform (LT) of nested analytic functions. To this aim, we provide a table of the first few values of the complete Bell's polynomials, which are then used to evaluate the LT of composite exponential functions. Furthermore a code for approximating the Laplace Transform of general analytic composite functions is created and presented. A graphical verification of the proposed technique is illustrated in the last section.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81741584","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 aims to develop a numerical approximation for the solution of the advection-diffusion equation with constant and variable coefficients. We propose a numerical solution for the equation associated with Robin's mixed boundary conditions perturbed with a small parameter $varepsilon$. The approximation is based on a couple of methods: A spectral method of Galerkin type with a basis composed from Legendre-polynomials and a Gauss quadrature of type Gauss-Lobatto applied for integral calculations with a stability and convergence analysis. In addition, a Crank-Nicolson scheme is used for temporal solution as a finite difference method. Several numerical examples are discussed to show the efficiency of the proposed numerical method, specially when $varepsilon$ tends to zero so that we obtain the exact solution of the classic problem with homogeneous Dirichlet boundary conditions. The numerical convergence is well presented in different examples. Therefore, we build an efficient numerical method for different types of partial differential equations with different boundary conditions.
{"title":"Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions","authors":"Zineb Laouar, N. Arar, Abdel-Fattah Talaat","doi":"10.31197/atnaa.1139533","DOIUrl":"https://doi.org/10.31197/atnaa.1139533","url":null,"abstract":"This paper aims to develop a numerical approximation for the solution of the advection-diffusion equation with constant and variable coefficients. We propose a numerical solution for the equation associated with Robin's mixed boundary conditions perturbed with a small parameter $varepsilon$. The approximation is based on a couple of methods: A spectral method of Galerkin type with a basis composed from Legendre-polynomials and a Gauss quadrature of type Gauss-Lobatto applied for integral calculations with a stability and convergence analysis. In addition, a Crank-Nicolson scheme is used for temporal solution as a finite difference method. Several numerical examples are discussed to show the efficiency of the proposed numerical method, specially when $varepsilon$ tends to zero so that we obtain the exact solution of the classic problem with homogeneous Dirichlet boundary conditions. The numerical convergence is well presented in different examples. Therefore, we build an efficient numerical method for different types of partial differential equations with different boundary conditions.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79597999","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 study a new nonlinear sequential differential prob- lem with nonlocal integral conditions that involve convergent series. The problem involves two fractional order operators: Riemann-Liouville inte- gral, Caputo and Riemann-Liouville derivatives. We prove an existence and uniqueness result. Also, we prove a new existence result. We end our paper by presenting some illustrative examples.
{"title":"A Sequential Differential Problem With Caputo and Riemann Liouville Derivatives Involving Convergent Series","authors":"Mahdi Rakah","doi":"10.31197/atnaa.1224234","DOIUrl":"https://doi.org/10.31197/atnaa.1224234","url":null,"abstract":"In this paper, we study a new nonlinear sequential differential prob- \u0000lem with nonlocal integral conditions that involve convergent series. The \u0000problem involves two fractional order operators: Riemann-Liouville inte- \u0000gral, Caputo and Riemann-Liouville derivatives. We prove an existence \u0000and uniqueness result. Also, we prove a new existence result. We end our \u0000paper by presenting some illustrative examples.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79501380","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}
Hallaci Ahmed, Professor DR., Krichen Bi̇lel, Mefteh Bi̇lel
This paper deals with the existence of weak solutions for an initial value problem involving Riemann-Liouville-type fractional derivatives. To this end, we transform the posed problem to a sum of two integral operators, then we apply a variant of Krasnoselskii’s fixed point theorem under weak topology to conclude the existence of weak solutions.
{"title":"New existence result under weak topology for fractional differential equations","authors":"Hallaci Ahmed, Professor DR., Krichen Bi̇lel, Mefteh Bi̇lel","doi":"10.31197/atnaa.1235476","DOIUrl":"https://doi.org/10.31197/atnaa.1235476","url":null,"abstract":"This paper deals with the existence of weak solutions for an initial value problem involving Riemann-Liouville-type fractional derivatives. To this end, we transform the posed problem to a sum of two integral operators, then we apply a variant of Krasnoselskii’s fixed point theorem under weak topology to conclude the existence of weak solutions.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85127091","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 (G'/G)-expansion method with the aid of symbolic computational system can be used to obtain the traveling wave solutions (hyperbolic, trigonometric and rational solutions) for nonlinear time-fractional evolution equations arising in mathematical physics and biology. In this work, we will process the analytical solutions of the time-fractional classical Boussinesq equation, the time-fractional Murray equation, and the space-time fractional Phi-four equation. With the fact that the method which we will propose in this paper is also a standard, direct and computerized method, the exact solutions for these equations are obtained.
{"title":"(G'/G)-EXPANSION METHOD TO SEEK TRAVELING WAVE SOLUTIONS FOR SOME FRACTIONAL NONLINEAR PDES ARISING IN NATURAL SCIENCES","authors":"Medjahed Djilali, Hakem Ali","doi":"10.31197/atnaa.1125691","DOIUrl":"https://doi.org/10.31197/atnaa.1125691","url":null,"abstract":"The (G'/G)-expansion method with the aid of symbolic computational system can be used to obtain the traveling wave solutions (hyperbolic, trigonometric and rational solutions) for nonlinear time-fractional evolution equations arising in mathematical physics and biology. In this work, we will process the analytical solutions of the time-fractional classical Boussinesq equation, the time-fractional Murray equation, and the space-time fractional Phi-four equation. With the fact that the method which we will propose in this paper is also a standard, direct and computerized method, the exact solutions for these equations are obtained.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76296127","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 new computational approach is presented to solve a boundary-value problem for a differential equation with piecewise constant argument of generalized type (DEPCAG). The presented technique is based on the Dzhumabaev parametrization method. A useful numerical algorithm is developed to obtain the numerical values from the problem. Numerical experiments are conducted to demonstrate the accuracy and efficiency.
{"title":"A new computational approach for solving a boundary-value problem for DEPCAG","authors":"Zh. M. Kadirbayeva, A. Assanova, E. Bakirova","doi":"10.31197/atnaa.1202501","DOIUrl":"https://doi.org/10.31197/atnaa.1202501","url":null,"abstract":"In this paper, a new computational approach is presented to solve a boundary-value problem for a differential equation with piecewise constant argument of generalized type (DEPCAG). The presented technique is based on the Dzhumabaev parametrization method. A useful numerical algorithm is developed to obtain the numerical values from the problem. Numerical experiments are conducted to demonstrate the accuracy and efficiency.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":"257 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77132820","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}
V. Mityushev, T. Gric, Z. Zhunussova, K. Dosmagulova
The $mathbb R$-linear boundary value problem in a multiply connected domain on a flat torus is considered. This problem is closely related to the Riemann-Hilbert problem on analytic functions. The considered problem arises in the homogenization procedure of random media with complex constants which express the permittivity of components. A new asymptotic formula for the effective permittivity tensor is derived. The formula contains location of inclusions in symbolic form. The application of the derived formula to investigation of the morphology of the tumor cells in disordered biological media is discussed.
{"title":"An asymptotic homogenization formula for complex permittivity and its application","authors":"V. Mityushev, T. Gric, Z. Zhunussova, K. Dosmagulova","doi":"10.31197/atnaa.1223064","DOIUrl":"https://doi.org/10.31197/atnaa.1223064","url":null,"abstract":"The $mathbb R$-linear boundary value problem in a multiply connected domain on a flat torus is considered. This problem is closely related to the Riemann-Hilbert problem on analytic functions. The considered problem arises in the homogenization procedure of random media with complex constants which express the permittivity of components. A new asymptotic formula for the effective permittivity tensor is derived. The formula contains location of inclusions in symbolic form. The application of the derived formula to investigation of the morphology of the tumor cells in disordered biological media is discussed.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":"97 2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91334311","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}
M. Jenaliyev, A. Kassymbekova, M. Yergaliyev, Bekzat Orynbasar
The work studies boundary value problems with non-dynamic and dynamic boundary conditions for one- and two-dimensional Boussinesq-type equations in domains representing a trapezoid, triangle, "curvilinear" trapezoid, "curvilinear" triangle, truncated cone, cone, truncated "curvilinear" cone, and "curvilinear" cone. Combining the methods of the theory of monotone operators and a priori estimates, in Sobolev classes, we have established theorems on the unique weak solvability of the boundary value problems under study.
{"title":"On boundary value problems for the Boussinesq-type equation with dynamic and non-dynamic boundary conditions","authors":"M. Jenaliyev, A. Kassymbekova, M. Yergaliyev, Bekzat Orynbasar","doi":"10.31197/atnaa.1215178","DOIUrl":"https://doi.org/10.31197/atnaa.1215178","url":null,"abstract":"The work studies boundary value problems with non-dynamic and dynamic boundary conditions for one- and two-dimensional Boussinesq-type equations in domains representing a trapezoid, triangle, \"curvilinear\" trapezoid, \"curvilinear\" triangle, truncated cone, cone, truncated \"curvilinear\" cone, and \"curvilinear\" cone. Combining the methods of the theory of monotone operators and a priori estimates, in Sobolev classes, we have established theorems on the unique weak solvability of the boundary value problems under study.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":"85 1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76070245","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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