Pub Date : 2021-06-01DOI: 10.32513/tmj/19322008124
Süleyman Çetinkaya, A. Demir
In this research, we discuss the construction of analytic solution of non-homogenous initial boundary value problem including PDEs of fractional order. By means of separation of variables method, the solution is constructed in the form of a Fourier series with respect to the eigenfunctions of a corresponding Sturm-Liouville eigenvalue problem including fractional derivative in Liouville-Caputo sense.
{"title":"Sequential time space fractional diffusion equation including nonhomogenous initial boundary conditions","authors":"Süleyman Çetinkaya, A. Demir","doi":"10.32513/tmj/19322008124","DOIUrl":"https://doi.org/10.32513/tmj/19322008124","url":null,"abstract":"In this research, we discuss the construction of analytic solution of non-homogenous initial boundary value problem including PDEs of fractional order. By means of separation of variables method, the solution is constructed in the form of a Fourier series with respect to the eigenfunctions of a corresponding Sturm-Liouville eigenvalue problem including fractional derivative in Liouville-Caputo sense.","PeriodicalId":43977,"journal":{"name":"Tbilisi Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44116235","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}
Pub Date : 2021-06-01DOI: 10.32513/tmj/19322008138
Merve Kara, Y. Yazlik
It is shown that the following $left( k+lright) $-order nonlinear difference equation $$x_{n}=frac{x_{n-k}x_{n-k-l}}{x_{n-l}left( a_{n}+b_{n}x_{n-k}x_{n-k-l}right)}, nin mathbb{N}_{0},$$ where $k,lin mathbb{N}$, $left(a_{n} right)_{nin mathbb{N}_{0}}$, $left(b_{n} right)_{nin mathbb{N}_{0}}$ and the initial values $x_{-i}$, $i=overline {1,k+l}$, are real numbers, can be solved and extended some results in literature. Also, by using obtained formulas, we give the forbidden set of the initial values for aforementioned equation and study the asymptotic behavior of well-defined solutions of above difference equation for the case $k=3$, $l=k$.
{"title":"Solvability of a $left( k+lright)$-order nonlinear difference equation","authors":"Merve Kara, Y. Yazlik","doi":"10.32513/tmj/19322008138","DOIUrl":"https://doi.org/10.32513/tmj/19322008138","url":null,"abstract":"It is shown that the following $left( k+lright) $-order nonlinear difference equation $$x_{n}=frac{x_{n-k}x_{n-k-l}}{x_{n-l}left( a_{n}+b_{n}x_{n-k}x_{n-k-l}right)}, nin mathbb{N}_{0},$$ where $k,lin mathbb{N}$, $left(a_{n} right)_{nin mathbb{N}_{0}}$, $left(b_{n} right)_{nin mathbb{N}_{0}}$ and the initial values $x_{-i}$, $i=overline {1,k+l}$, are real numbers, can be solved and extended some results in literature. Also, by using obtained formulas, we give the forbidden set of the initial values for aforementioned equation and study the asymptotic behavior of well-defined solutions of above difference equation for the case $k=3$, $l=k$.","PeriodicalId":43977,"journal":{"name":"Tbilisi Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48366836","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}
Pub Date : 2021-06-01DOI: 10.32513/tmj/19322008123
Nilay Sager, B. Sağır
In this work, we construct vector spaces $l_{p}left( mathbb{BC}left(Nright) right) $ of absolutely $p-$ summable $ast -$bicomplex sequences with the $ast -$ norm $overset{..}{parallel }.overset{..}{parallel }_{2,l_{p}left( mathbb{BC}left( Nright) right) }$ over the field $mathbb{C}left( Nright).$ Also, we show that some inclusion relations hold and these vector spaces are Banach spaces by using Minkowski's inequality in $mathbb{BC}left( Nright) $ with respect to $overset{..}{parallel }.overset{..}{parallel }_{2}.$
{"title":"Banach spaces $l_{p}left( mathbb{BC}left( Nright) right) $ with the $ast -$ norm $overset{..}{parallel }.overset{..}{parallel }_{2,l_{p}left( mathbb{BC}left( Nright) right) }$ and some properties","authors":"Nilay Sager, B. Sağır","doi":"10.32513/tmj/19322008123","DOIUrl":"https://doi.org/10.32513/tmj/19322008123","url":null,"abstract":"In this work, we construct vector spaces $l_{p}left( mathbb{BC}left(Nright) right) $ of absolutely $p-$ summable $ast -$bicomplex sequences with the $ast -$ norm $overset{..}{parallel }.overset{..}{parallel }_{2,l_{p}left( mathbb{BC}left( Nright) right) }$ over the field $mathbb{C}left( Nright).$ Also, we show that some inclusion relations hold and these vector spaces are Banach spaces by using Minkowski's inequality in $mathbb{BC}left( Nright) $ with respect to $overset{..}{parallel }.overset{..}{parallel }_{2}.$","PeriodicalId":43977,"journal":{"name":"Tbilisi Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42693351","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}
Pub Date : 2021-06-01DOI: 10.32513/tmj/19322008135
S. Boughaba, A. Boussayoud, N. Saba, K. Kanuri
In this study, we introduce a new family of generating functions of products of bivariate complex Fibonacci polynomials with Gaussian Fibonacci numbers, Gaussian Lucas numbers, Gaussian Jacobsthal numbers, Gaussian Jacobsthal Lucas numbers, Gaussian Pell numbers and Gaussian Pell Lucas numbers.
{"title":"A new family of generating functions of binary products of bivariate complex Fibonacci polynomials and Gaussian numbers","authors":"S. Boughaba, A. Boussayoud, N. Saba, K. Kanuri","doi":"10.32513/tmj/19322008135","DOIUrl":"https://doi.org/10.32513/tmj/19322008135","url":null,"abstract":"In this study, we introduce a new family of generating functions of products of bivariate complex Fibonacci polynomials with Gaussian Fibonacci numbers, Gaussian Lucas numbers, Gaussian Jacobsthal numbers, Gaussian Jacobsthal Lucas numbers, Gaussian Pell numbers and Gaussian Pell Lucas numbers.","PeriodicalId":43977,"journal":{"name":"Tbilisi Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47901389","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}
Pub Date : 2021-06-01DOI: 10.32513/tmj/19322008133
A. Oğuz, S. Topal
This paper is concerned with the existence and nonexistence of positive solutions for a system of nonlinear second order dynamic equations with multi-point boundary conditions on time scales.
研究了一类时间尺度上具有多点边界条件的二阶非线性动力方程组正解的存在性和不存在性。
{"title":"On a system of second-order multi-point boundary value problems on time scales","authors":"A. Oğuz, S. Topal","doi":"10.32513/tmj/19322008133","DOIUrl":"https://doi.org/10.32513/tmj/19322008133","url":null,"abstract":"This paper is concerned with the existence and nonexistence of positive solutions for a system of nonlinear second order dynamic equations with multi-point boundary conditions on time scales.","PeriodicalId":43977,"journal":{"name":"Tbilisi Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48458697","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}
Pub Date : 2021-06-01DOI: 10.32513/tmj/19322008125
Sevda Akdağ, P. Mathur
In this paper, considering the concept of $f-$statistical convergence which is a generalization of statistical convergence and is intermediate between the ordinary convergence and the statistical convergence, we obtain a $f-$statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued $B-$continuous functions on a compact subset of the real line. Furthermore, we compute the rates of $f-$statistical convergence with the help mixed modulus of smoothness.
{"title":"$f-$Statistical approximation to Bögel-type continuous functions","authors":"Sevda Akdağ, P. Mathur","doi":"10.32513/tmj/19322008125","DOIUrl":"https://doi.org/10.32513/tmj/19322008125","url":null,"abstract":"In this paper, considering the concept of $f-$statistical convergence which is a generalization of statistical convergence and is intermediate between the ordinary convergence and the statistical convergence, we obtain a $f-$statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued $B-$continuous functions on a compact subset of the real line. Furthermore, we compute the rates of $f-$statistical convergence with the help mixed modulus of smoothness.","PeriodicalId":43977,"journal":{"name":"Tbilisi Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47990237","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}
Pub Date : 2021-06-01DOI: 10.32513/tmj/19322008136
A. Aghili
In this article, the author used Mellin and Kontorovich-Lebedev transforms to establish certain integrals involving Macdonald's functions. Transform method is a powerful tool for solving singular integral equations, evaluation of certain integrals and solution to partial differential equations. The result reveals that the transform method is very convenient and effective.
{"title":"Some identities for Mellin, Kontorovich-Lebedev transforms with applications","authors":"A. Aghili","doi":"10.32513/tmj/19322008136","DOIUrl":"https://doi.org/10.32513/tmj/19322008136","url":null,"abstract":"In this article, the author used Mellin and Kontorovich-Lebedev transforms to establish certain integrals involving Macdonald's functions. Transform method is a powerful tool for solving singular integral equations, evaluation of certain integrals and solution to partial differential equations. The result reveals that the transform method is very convenient and effective.","PeriodicalId":43977,"journal":{"name":"Tbilisi Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45764101","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}
Pub Date : 2021-06-01DOI: 10.32513/tmj/19322008128
Gyan Prakash Rathore, Anupma Rastogi
After the recent works of Biswas [19], on the idea of relative $(p, q)$-th order and relative $(p, q)$-th type of composite $p$-adic entire functions, in this paper we establish some results of growth properties of an $p$-adic entire functions of several complex variables with respect to another $p$-adic entire function of several complex variables $gin mathcal{A}(mathbb{K})$ on the basis of their relative $(p, q)$-th order, relative $(p, q)$-th lower order, relative $(p,q)$-th type, relative $(p,q)$-th type of an entire functions of several complex variables.
{"title":"On the growth properties of relative $(p, q)$-th order and relative $(p, q)$-th type of composite $p$-adic entire functions of several complex variables","authors":"Gyan Prakash Rathore, Anupma Rastogi","doi":"10.32513/tmj/19322008128","DOIUrl":"https://doi.org/10.32513/tmj/19322008128","url":null,"abstract":"After the recent works of Biswas [19], on the idea of relative $(p, q)$-th order and relative $(p, q)$-th type of composite $p$-adic entire functions, in this paper we establish some results of growth properties of an $p$-adic entire functions of several complex variables with respect to another $p$-adic entire function of several complex variables $gin mathcal{A}(mathbb{K})$ on the basis of their relative $(p, q)$-th order, relative $(p, q)$-th lower order, relative $(p,q)$-th type, relative $(p,q)$-th type of an entire functions of several complex variables.","PeriodicalId":43977,"journal":{"name":"Tbilisi Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43808025","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}
Pub Date : 2021-06-01DOI: 10.32513/tmj/19322008118
Xingguo Liu, Xuehua Yang, Haixiang Zhang, Yanling Liu
This paper studies the fourth-order problem with multi-term time fractional integral operator under simply supported type conditions. We first introduce a novel computational approach, the discrete singular convolution (DSC) algorithm, for analyzing this problem. Detailed discrete formulations and the treatment of simply supported boundary condition are established. We provide some numerical results to demonstrate the validity and applicability of the proposed technique. Comprehensive comparisons are given based on a variety of time increment, grid spacing and wave number. Unified features of the DSC algorithm for solving differential equations are explored. It is demonstrated that the DSC algorithm is an accurate, stable and robust approach for solving the fourth-order integro-differential equation with multi-term time fractional integral operator.
{"title":"Discrete singular convolution for fourth-order multi-term time fractional equation","authors":"Xingguo Liu, Xuehua Yang, Haixiang Zhang, Yanling Liu","doi":"10.32513/tmj/19322008118","DOIUrl":"https://doi.org/10.32513/tmj/19322008118","url":null,"abstract":"This paper studies the fourth-order problem with multi-term time fractional integral operator under simply supported type conditions. We first introduce a novel computational approach, the discrete singular convolution (DSC) algorithm, for analyzing this problem. Detailed discrete formulations and the treatment of simply supported boundary condition are established. We provide some numerical results to demonstrate the validity and applicability of the proposed technique. Comprehensive comparisons are given based on a variety of time increment, grid spacing and wave number. Unified features of the DSC algorithm for solving differential equations are explored. It is demonstrated that the DSC algorithm is an accurate, stable and robust approach for solving the fourth-order integro-differential equation with multi-term time fractional integral operator.","PeriodicalId":43977,"journal":{"name":"Tbilisi Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48639106","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}
Pub Date : 2021-06-01DOI: 10.32513/tmj/19322008120
S. B. G. Karakoç, Khalid K. Ali
This paper aims to obtain exact and numerical solutions of the nonlinear Benjamin Bona Mahony-Burgers (BBM-Burgers) equation. Here, we propose the modified Kudryashov method for getting the exact traveling wave solutions of BBM-Burgers equation and a septic B-spline collocation finite element method for numerical investigations. The numerical method is validated by studying solitary wave motion. Linear stability analysis of the numerical scheme is done with Fourier method based on von-Neumann theory. To show suitability and robustness of the new numerical algorithm, error norms $L_{2}$, $L_{infty }$ and three invariants $I_{1},I_{2}$ and $I_{3}$ are calculated and obtained results are given both numerically and graphically. The obtained results state that our exact and numerical schemes ensure evident and they are penetrative mathematical instruments for solving nonlinear evolution equation.
本文旨在获得非线性Benjamin Bona Mahony-Burgers (BBM-Burgers)方程的精确数值解。本文提出了修正Kudryashov法求解BBM-Burgers方程的行波精确解，并提出了一种b样条配点法进行数值研究。通过对孤立波动的研究，对数值方法进行了验证。采用基于冯-诺伊曼理论的傅里叶方法对数值格式进行了线性稳定性分析。为了证明新数值算法的适用性和鲁棒性，计算了误差范数$L_{2}$、$L_{infty }$和三个不变量$I_{1},I_{2}$、$I_{3}$，并给出了数值和图形结果。得到的结果表明，我们的精确格式和数值格式是明显的，是求解非线性演化方程的渗透性数学工具。
{"title":"Theoretical and computational structures on solitary wave solutions of Benjamin Bona Mahony-Burgers equation","authors":"S. B. G. Karakoç, Khalid K. Ali","doi":"10.32513/tmj/19322008120","DOIUrl":"https://doi.org/10.32513/tmj/19322008120","url":null,"abstract":"This paper aims to obtain exact and numerical solutions of the nonlinear Benjamin Bona Mahony-Burgers (BBM-Burgers) equation. Here, we propose the modified Kudryashov method for getting the exact traveling wave solutions of BBM-Burgers equation and a septic B-spline collocation finite element method for numerical investigations. The numerical method is validated by studying solitary wave motion. Linear stability analysis of the numerical scheme is done with Fourier method based on von-Neumann theory. To show suitability and robustness of the new numerical algorithm, error norms $L_{2}$, $L_{infty }$ and three invariants $I_{1},I_{2}$ and $I_{3}$ are calculated and obtained results are given both numerically and graphically. The obtained results state that our exact and numerical schemes ensure evident and they are penetrative mathematical instruments for solving nonlinear evolution equation.","PeriodicalId":43977,"journal":{"name":"Tbilisi Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47290290","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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