This computational study investigates a class of singularly perturbed second-order boundary-value problems having dual (twin) boundary layers and simple turning points. It is well-known that the classical discretization methods fail to resolve sharp gradients arising in solving singularly perturbed differential equations as the perturbation (diffusion) parameter decreases, i.e., ε → 0+. To this end, this paper proposes a semi-analytic hybrid method consisting of a numerical procedure based on finite differences and an asymptotic method called the Successive Complementary Expansion Method (SCEM) to approximate the solution of such problems. Two numerical experiments are provided to demonstrate the method’s implementation and to evaluate its computational performance. Several comparisons with the numerical results existing in the literature are also made. The numerical observations reveal that the hybrid method leads to good solution profiles and achieves this in only a few iterations.
{"title":"A Semi-analytic method for solving singularly perturbed twin-Layer Problems with a turning Point","authors":"Süleyman Cengizci, D. Kumar, M. Atay","doi":"10.3846/mma.2023.14953","DOIUrl":"https://doi.org/10.3846/mma.2023.14953","url":null,"abstract":"This computational study investigates a class of singularly perturbed second-order boundary-value problems having dual (twin) boundary layers and simple turning points. It is well-known that the classical discretization methods fail to resolve sharp gradients arising in solving singularly perturbed differential equations as the perturbation (diffusion) parameter decreases, i.e., ε → 0+. To this end, this paper proposes a semi-analytic hybrid method consisting of a numerical procedure based on finite differences and an asymptotic method called the Successive Complementary Expansion Method (SCEM) to approximate the solution of such problems. Two numerical experiments are provided to demonstrate the method’s implementation and to evaluate its computational performance. Several comparisons with the numerical results existing in the literature are also made. The numerical observations reveal that the hybrid method leads to good solution profiles and achieves this in only a few iterations.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":"65 1","pages":"102-117"},"PeriodicalIF":1.8,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84427297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Construction of second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods with Runge–Kutta stability property requires to generate the corresponding conditions depending of the parameters of the methods. These conditions which are a system of polynomial equations can not be produced by symbolic manipulation packages for the methods of order p ≥ 5. In this paper, we describe an approach to construct SDIMSIMs with Runge–Kutta stability property by using some variant of the Fourier series method which has been already used for the construction of high order general linear methods. Examples of explicit and implicit SDIMSIMs of order five and six are given which respectively are appropriate for both non-stiff and stiff differential systems in a sequential computing environment. Finally, the efficiency of the constructed methods is verified by providing some numerical experiments.
{"title":"High order second derivative diagonally Implicit Multistage Integration Methods for ODEs","authors":"M. Sharifi, A. Abdi, M. Braś, G. Hojjati","doi":"10.3846/mma.2023.16102","DOIUrl":"https://doi.org/10.3846/mma.2023.16102","url":null,"abstract":"Construction of second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods with Runge–Kutta stability property requires to generate the corresponding conditions depending of the parameters of the methods. These conditions which are a system of polynomial equations can not be produced by symbolic manipulation packages for the methods of order p ≥ 5. In this paper, we describe an approach to construct SDIMSIMs with Runge–Kutta stability property by using some variant of the Fourier series method which has been already used for the construction of high order general linear methods. Examples of explicit and implicit SDIMSIMs of order five and six are given which respectively are appropriate for both non-stiff and stiff differential systems in a sequential computing environment. Finally, the efficiency of the constructed methods is verified by providing some numerical experiments.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":"16 1","pages":"53-70"},"PeriodicalIF":1.8,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88105686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The goal of the present paper is to study the viscoelastic wave equation with the time-varying delay under initial-boundary value conditions. By using the multiplier method together with some properties of the convex functions, the explicit and general stability results of the total energy are proved under the general assumption on the relaxation function g.
{"title":"Asymptotic stability for a viscoelastic equation with the Time-varying delay","authors":"Menglan Liao, Zhong Tan","doi":"10.3846/mma.2023.16160","DOIUrl":"https://doi.org/10.3846/mma.2023.16160","url":null,"abstract":"The goal of the present paper is to study the viscoelastic wave equation with the time-varying delay under initial-boundary value conditions. By using the multiplier method together with some properties of the convex functions, the explicit and general stability results of the total energy are proved under the general assumption on the relaxation function g.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":"3 1","pages":"23-41"},"PeriodicalIF":1.8,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79233372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider and investigate an altering points problem involving generalized accretive mappings over closed convex subsets of a real uniformly smooth Banach space. Parallel Mann and parallel S-iterative methods are suggested to analyze the approximate solution of altering points problem. Consequently, some systems of generalized variational inclusions and generalized variational inequalities are also explored using the conceptual framework of altering points. Convergence of suggested iterative methods are verified by an illustrative numerical example.
{"title":"Approximation of iterative Methods for altering Points Problem with Applications","authors":"Aysha Khan, M. Akram, M. Dilshad","doi":"10.3846/mma.2023.14858","DOIUrl":"https://doi.org/10.3846/mma.2023.14858","url":null,"abstract":"In this paper, we consider and investigate an altering points problem involving generalized accretive mappings over closed convex subsets of a real uniformly smooth Banach space. Parallel Mann and parallel S-iterative methods are suggested to analyze the approximate solution of altering points problem. Consequently, some systems of generalized variational inclusions and generalized variational inequalities are also explored using the conceptual framework of altering points. Convergence of suggested iterative methods are verified by an illustrative numerical example.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":"547 ","pages":"118-145"},"PeriodicalIF":1.8,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72424589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The consistency of the classical Richardson extrapolation (CRE), a simple and robust computational device, is analysed for the case where the underlying method is an explicit one-step numerical method for ordinary differential equations with order of consistency one or two. It is shown in the classical framework that the CRE increases the order of consistency by one. The convergence of the method is proved by the assumption that the time-stepping operator of the base method has the Lipschitz property in its second argument.
{"title":"On the Consistency and convergence of Classical Richardson extrapolation as Applied to Explicit One-Step Methods","authors":"Teshome Bayleyegn, I. Faragó, Ágnes Havasi","doi":"10.3846/mma.2023.16283","DOIUrl":"https://doi.org/10.3846/mma.2023.16283","url":null,"abstract":"The consistency of the classical Richardson extrapolation (CRE), a simple and robust computational device, is analysed for the case where the underlying method is an explicit one-step numerical method for ordinary differential equations with order of consistency one or two. It is shown in the classical framework that the CRE increases the order of consistency by one. The convergence of the method is proved by the assumption that the time-stepping operator of the base method has the Lipschitz property in its second argument.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":"13 1","pages":"42-52"},"PeriodicalIF":1.8,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74548779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this research article Hermite wavelet based Galerkin method is developed for the numerical solution of Volterra integro-differential equations in onedimension with initial and boundary conditions. These equations include the partial differential of an unknown function and the integral term containing the unknown function which is the memory of the problem. Wavelet analysis is a recently developed mathematical tool in applied mathematics. For this purpose, Hermit wavelet Galerkin method has proven a very powerful numerical technique for the stable and accurate solution of giving boundary value problem. The theorem of convergence analysis and compare some numerical examples with the use of the proposed method and the exact solutions shows the efficiency and high accuracy of the proposed method. Several figures are plotted to establish the error analysis of the approach presented.
{"title":"An Effective Computational Approach based on Hermite wavelet Galerkin for solving parabolic Volterra Partial integro differential equations and its convergence Analysis","authors":"Yaser Rostami","doi":"10.3846/mma.2023.15690","DOIUrl":"https://doi.org/10.3846/mma.2023.15690","url":null,"abstract":"In this research article Hermite wavelet based Galerkin method is developed for the numerical solution of Volterra integro-differential equations in onedimension with initial and boundary conditions. These equations include the partial differential of an unknown function and the integral term containing the unknown function which is the memory of the problem. Wavelet analysis is a recently developed mathematical tool in applied mathematics. For this purpose, Hermit wavelet Galerkin method has proven a very powerful numerical technique for the stable and accurate solution of giving boundary value problem. The theorem of convergence analysis and compare some numerical examples with the use of the proposed method and the exact solutions shows the efficiency and high accuracy of the proposed method. Several figures are plotted to establish the error analysis of the approach presented.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":"27 1","pages":"163-179"},"PeriodicalIF":1.8,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82948102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the paper, we construct an absolutely convergent Dirichlet series which in the mean is close to the periodic Hurwitz zeta-function, and has the universality property on the approximation of a wide class of analytic functions.
{"title":"On a Dirichlet Series Connected to a periodic Hurwitz zeta-function with Transcendental and rational parameter","authors":"A. Balčiūnas, A. Laurinčikas, M. Stoncelis","doi":"10.3846/mma.2023.17222","DOIUrl":"https://doi.org/10.3846/mma.2023.17222","url":null,"abstract":"In the paper, we construct an absolutely convergent Dirichlet series which in the mean is close to the periodic Hurwitz zeta-function, and has the universality property on the approximation of a wide class of analytic functions.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":"17 1","pages":"91-101"},"PeriodicalIF":1.8,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82971745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, two multi-step iterative techniques of fifth order convergence are explored to solve nonlinear equations. The techniques are designed with the prime objective of keeping the computational cost as low as possible. To claim this objective, the efficiency indices are determined and compared with the efficiencies of the existing techniques of same order. The outcome of comparison analysis is remarkable from the view of high computational efficiency of new methods. Performance and stability are illustrated by executing the numerical tests on some nonlinear problems of diverse nature. The entire analysis significantly favors the new techniques compared to their existing counterparts, especially for the case of large dimensional systems.
{"title":"Simple and Efficient Fifth order Solvers for Systems of nonlinear Problems","authors":"Harmandeep Singh, J. Sharma","doi":"10.3846/mma.2023.16244","DOIUrl":"https://doi.org/10.3846/mma.2023.16244","url":null,"abstract":"In this study, two multi-step iterative techniques of fifth order convergence are explored to solve nonlinear equations. The techniques are designed with the prime objective of keeping the computational cost as low as possible. To claim this objective, the efficiency indices are determined and compared with the efficiencies of the existing techniques of same order. The outcome of comparison analysis is remarkable from the view of high computational efficiency of new methods. Performance and stability are illustrated by executing the numerical tests on some nonlinear problems of diverse nature. The entire analysis significantly favors the new techniques compared to their existing counterparts, especially for the case of large dimensional systems.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":"64 1","pages":"1-22"},"PeriodicalIF":1.8,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91109542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, images of the tempered Ψ-Hilfer fractional integral and the tempered Ψ-Caputo fractional derivative under the generalized Laplace transform are derived. The results are applied to find a solution to an initial value problem for a nonhomogeneous linear fractional differential equation with the tempered Ψ-Caputo fractional derivative of an order α for n− 1 <α
{"title":"Generalized Laplace transform and tempered ψ-Caputo fractional derivative","authors":"M. Medved', M. Pospíšil","doi":"10.3846/mma.2023.16370","DOIUrl":"https://doi.org/10.3846/mma.2023.16370","url":null,"abstract":"In this paper, images of the tempered Ψ-Hilfer fractional integral and the tempered Ψ-Caputo fractional derivative under the generalized Laplace transform are derived. The results are applied to find a solution to an initial value problem for a nonhomogeneous linear fractional differential equation with the tempered Ψ-Caputo fractional derivative of an order α for n− 1 <α<n N. An illustrative example is given for 0 <α<1 comparing solutions to the same initial value problem but with different tempering and Ψ.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":"21 1","pages":"146-162"},"PeriodicalIF":1.8,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91140454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The integral model with finite memory is employed to analyze the timeline of COVID-19 epidemic in the United Kingdom and government actions to mitigate it. The model uses a realistic infection distribution. The time-varying transmission rate is determined from Volterra integral equation of the first kind. The authors construct and justify an efficient regularization algorithm for finding the transmission rate. The model and algorithm are approbated on the UK data with several waves of COVID-19 and demonstrate a remarkable resemblance between real and simulated dynamics. The timing of government preventive measures and their impact on the epidemic dynamics are discussed.
{"title":"Integral Model of CoViD-19 spread and Mitigation in UK: identification of Transmission rate","authors":"N. Hritonenko, Caroline Satsky, Y. Yatsenko","doi":"10.3846/mma.2022.15708","DOIUrl":"https://doi.org/10.3846/mma.2022.15708","url":null,"abstract":"The integral model with finite memory is employed to analyze the timeline of COVID-19 epidemic in the United Kingdom and government actions to mitigate it. The model uses a realistic infection distribution. The time-varying transmission rate is determined from Volterra integral equation of the first kind. The authors construct and justify an efficient regularization algorithm for finding the transmission rate. The model and algorithm are approbated on the UK data with several waves of COVID-19 and demonstrate a remarkable resemblance between real and simulated dynamics. The timing of government preventive measures and their impact on the epidemic dynamics are discussed.","PeriodicalId":49861,"journal":{"name":"Mathematical Modelling and Analysis","volume":"186 1","pages":"573-589"},"PeriodicalIF":1.8,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89003224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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