Pub Date : 2021-09-25DOI: 10.52280/pujm.2021.530904
Hameed Ullah Jan
In engineering and mathematical physics nonlinear evolutionary equations play an important role. Kawahara equation is one of the famous nonlinear evolution equation appeared in the theories of shallow water waves possessing surface tension, capillary-gravity waves and also magneto-acoustic waves in a plasma. Another specific subjective parts of arrangements for some of evolution equations demonstrated by investigations, which connect alongwith their large-time behavior named as eventual time periodicity uncovered across solutions to IBVPs (initialboundary-value problems). In this study eventual periodicity of solutions for the generalized fifth order Kawahara equation (IBVP) on bounded domain coupled with periodic boundary condition will explored numerically utilizing meshless technique called as Radial basis function generated finite difference (RBF-FD) method.
{"title":"Approximation and Eventual periodicity of Generalized Kawahara equation using\u0000RBF-FD method","authors":"Hameed Ullah Jan","doi":"10.52280/pujm.2021.530904","DOIUrl":"https://doi.org/10.52280/pujm.2021.530904","url":null,"abstract":"In engineering and mathematical physics nonlinear evolutionary equations play an important role. Kawahara equation is one of the famous nonlinear evolution equation appeared in the theories of shallow water waves possessing surface tension, capillary-gravity waves and also magneto-acoustic waves in a plasma. Another specific subjective parts of arrangements for some of evolution equations demonstrated by investigations, which connect alongwith their large-time behavior named as eventual time periodicity uncovered across solutions to IBVPs (initialboundary-value problems). In this study eventual periodicity of solutions for the generalized fifth order Kawahara equation (IBVP) on bounded domain coupled with periodic boundary condition will explored numerically\u0000utilizing meshless technique called as Radial basis function generated finite difference (RBF-FD) method.","PeriodicalId":205373,"journal":{"name":"Punjab University Journal of Mathematics","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123671967","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 : 2021-09-25DOI: 10.52280/pujm.2021.530903
Decision-making is one of the contemporary issues in this modern era due to the interaction of risk and uncertainties in every aspect of the daily lives of human beings. Accordingly, solving practical optimization problems tends to be more challenging. The fundamental reason for this investigation is to explore an effective solution technique for multi-objective optimization problems (MOOPs) in an intuitionistic fuzzy environment (IFE) addressing the issue of determining proper violation parameters and tolerances to the objectives and constraints. The other significant characteristic of this study is the consideration of the decisionmaker’s perspective, namely, optimistic, pessimistic and mixed views in the solution procedure. In the proposed method, compared to the existing study, the required number of iterations and stages are considerably reduced in solving intuitionistic fuzzy multi-objective optimization problems (IFMOOPs). Hence it has imperative advantages in solving complex real-world problems without much difficulty. One problem is solved to demonstrate the competency of the planned approach. A comparative analysis is also undertaken to ascertain the efficiency of the technique.
{"title":"http://pu.edu.pk/images/journal/maths/PDF/Paper_3_53_9_2021.pdf","authors":"","doi":"10.52280/pujm.2021.530903","DOIUrl":"https://doi.org/10.52280/pujm.2021.530903","url":null,"abstract":"Decision-making is one of the contemporary issues in this modern era due to the interaction of risk and uncertainties in every aspect of the daily lives of human beings. Accordingly, solving practical optimization problems tends to be more challenging. The fundamental reason for this investigation is to explore an effective solution technique\u0000for multi-objective optimization problems (MOOPs) in an intuitionistic fuzzy environment (IFE) addressing the issue of determining proper violation parameters and tolerances to the objectives and constraints. The other significant characteristic of this study is the consideration of the decisionmaker’s perspective, namely, optimistic, pessimistic and mixed views in the solution procedure. In the proposed method, compared to the existing study, the required number of iterations and stages are considerably reduced in solving intuitionistic fuzzy multi-objective optimization problems (IFMOOPs). Hence it has imperative advantages in solving complex real-world problems without much difficulty. One problem is solved to demonstrate the competency of the planned approach. A comparative analysis is also undertaken to ascertain the efficiency of the technique.","PeriodicalId":205373,"journal":{"name":"Punjab University Journal of Mathematics","volume":"231 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125819924","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 : 2021-09-25DOI: 10.52280/pujm.2021.530902
Muhammad Ihsan, M. Saeed, Atiqe Ur Rahman
Soft set theory is considered as the preeminent tool to tackle the problems involving vagueness by controlling all complexities of optimization theory, fuzzy set theory and interval theory. Some models have been developed to solve problems in decision making and medical diagnosis with one expert by using this theory. This causes a problem with those who use questionnaires in their research. Soft expert set overcomes this problem and facilitates the user to know the opinion of all experts in one model. The concept of convexity plays a key role to deal optimization, pattern recognition-classification and many other related topics in operation research, numerical analysis and other disciplines of mathematical sciences. In this study, a mathematical cum abstract technique is employed to develop basic concept of convex and concave soft expert sets to deal with their important applications. Some classical results on convexity cum concavity are modified under uncertain multi-decisive environment with the support of explicatory proofs.
{"title":"A Rudimentary Approach to Develop Context for Convexity cum Concavity on Soft Expert Set with Some Generalized Results","authors":"Muhammad Ihsan, M. Saeed, Atiqe Ur Rahman","doi":"10.52280/pujm.2021.530902","DOIUrl":"https://doi.org/10.52280/pujm.2021.530902","url":null,"abstract":"Soft set theory is considered as the preeminent tool to tackle the problems involving vagueness by controlling all complexities of optimization theory, fuzzy set theory and interval theory. Some models have been developed to solve problems in decision making and medical diagnosis with one expert by using this theory. This causes a problem with those who use questionnaires in their research. Soft expert set overcomes this problem and facilitates the user to know the opinion of all experts in one model. The concept of convexity plays a key role to deal optimization, pattern recognition-classification and many other related topics in operation research, numerical analysis and other disciplines of mathematical sciences. In this study, a mathematical cum abstract technique is employed to develop basic concept of convex and concave soft expert sets to deal with their important applications. Some classical results on convexity cum concavity are modified under uncertain multi-decisive environment with the support of explicatory proofs.","PeriodicalId":205373,"journal":{"name":"Punjab University Journal of Mathematics","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127593580","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 : 2021-09-25DOI: 10.52280/pujm.2021.530905
Sleyman enyurt, K. Eren
In this article, we investigate the regular Smarandache curves constructed from the Frenet vectors of spacelike Salkowski curve with a timelike principal normal. In the first part of the study, literature research was conducted. In the second part, general information about the curve and spacelike Salkowski curve in Minkowski space are given. In the last part, the Frenet apparatus of the Smarandache curves are calculated. We draw a graphic of the obtained Smarandache curves and some related results about Smarandache curves are given.
{"title":"Some Smarandache Curves Constructed from a Spacelike Salkowski Curve with\u0000Timelike Principal Normal","authors":"Sleyman enyurt, K. Eren","doi":"10.52280/pujm.2021.530905","DOIUrl":"https://doi.org/10.52280/pujm.2021.530905","url":null,"abstract":"In this article, we investigate the regular Smarandache curves constructed from the Frenet vectors of spacelike Salkowski curve with a timelike principal normal. In the first part of the study, literature research was conducted. In the second part, general information about the curve and spacelike Salkowski curve in Minkowski space are given. In the last\u0000part, the Frenet apparatus of the Smarandache curves are calculated. We draw a graphic of the obtained Smarandache curves and some related results about Smarandache curves are given.","PeriodicalId":205373,"journal":{"name":"Punjab University Journal of Mathematics","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114652801","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 : 2021-08-26DOI: 10.52280/pujm.2021.530802
Md. Shafiqul Islam, S. Bhowmick
In this article, a new recurrence relation formula for Bernoulli numbers have been derived, and sum of integer exponents of natural numbers has been revisited from this novel perspective. Some interesting pedagogical experiments on wording and presentation of mathematical derivation have been attempted, and development from first principle have been undertaken in line with this experimental approach.
{"title":"Linear Algebraic Approach to Formulate A New Recurrence Relation for Bernoulli\u0000Numbers from the Power-Sum of Natural Numbers with Experiments on Pedagogy","authors":"Md. Shafiqul Islam, S. Bhowmick","doi":"10.52280/pujm.2021.530802","DOIUrl":"https://doi.org/10.52280/pujm.2021.530802","url":null,"abstract":"In this article, a new recurrence relation formula for Bernoulli\u0000numbers have been derived, and sum of integer exponents of natural numbers has been revisited from this novel perspective. Some interesting pedagogical experiments on wording and presentation of mathematical derivation have been attempted, and development from first principle have been\u0000undertaken in line with this experimental approach.","PeriodicalId":205373,"journal":{"name":"Punjab University Journal of Mathematics","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123120649","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 : 2021-08-26DOI: 10.52280/pujm.2021.530805
I. Zari, Jinnah
This research article is concerned with the solution of hydrodynamic stability based linear and nonlinear fourteenth order differential problem, which has great significance in applied physics, astrophysics, applied mathematics, engineering departments. The homotopy perturbation method (HPM) and optimal homotopy asymptotic method (OHAM) are applied for the solution of the existed problem. These semi analytical techniques are continuously evolved to solve diverse range of linear and nonlinear problems with effective approximate agents which is a rapid approach to the exact solutions. This approach is effectively proposed with different numerical examples, which are taken from literature. Numerical results are accomplished by phrase of convergent series solutions and approach to the accurate solutions only by taking minimum steps. The numerical results are exercised with exact solutions, cubic polynomial spline technique (CPST) and cubic non-polynomial spline technique (CNPST), excellent agreement has been observed. The observations suggested that OHAM and HPM performed excellent in comparison to the CPST and CNPST in terms of solution, which demonstrated the effectiveness, potential and validity of suggested schemes in reality and acquired results are of top-level perfection.
{"title":"Semi-analytical solutions for the hydrodynamic stability based nonlinear fourteenth\u0000order differential problem","authors":"I. Zari, Jinnah","doi":"10.52280/pujm.2021.530805","DOIUrl":"https://doi.org/10.52280/pujm.2021.530805","url":null,"abstract":"This research article is concerned with the solution of hydrodynamic stability based linear and nonlinear fourteenth order differential\u0000problem, which has great significance in applied physics, astrophysics,\u0000applied mathematics, engineering departments. The homotopy perturbation method (HPM) and optimal homotopy asymptotic method (OHAM)\u0000are applied for the solution of the existed problem. These semi analytical\u0000techniques are continuously evolved to solve diverse range of linear and\u0000nonlinear problems with effective approximate agents which is a rapid approach to the exact solutions. This approach is effectively proposed with\u0000different numerical examples, which are taken from literature. Numerical results are accomplished by phrase of convergent series solutions and\u0000approach to the accurate solutions only by taking minimum steps. The numerical results are exercised with exact solutions, cubic polynomial spline\u0000technique (CPST) and cubic non-polynomial spline technique (CNPST),\u0000excellent agreement has been observed. The observations suggested that\u0000OHAM and HPM performed excellent in comparison to the CPST and\u0000CNPST in terms of solution, which demonstrated the effectiveness, potential and validity of suggested schemes in reality and acquired results\u0000are of top-level perfection.","PeriodicalId":205373,"journal":{"name":"Punjab University Journal of Mathematics","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133985316","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 : 2021-08-26DOI: 10.52280/pujm.2021.530801
Let p be a prime greater than or equal to 5. In this paper, by using the harmonic numbers and Fermat quotient we establish congruences involving the sums p−1 X2 k=1 µ k r ¶ Hk, p−1 X2 k=1 ¡ 2k k ¢2 16k H (2) k and p−1 X2 k=1 1 4 k µ 2k k ¶ H (3) k . For example, p−1 X2 k=0 ¡ 2k k ¢2 16k H (2) k ≡ 4E2p−4 − 8Ep−3 ¡ mod p 2 ¢ , where H (m) k are the generalized harmonic numbers of order m and En are Euler numbers
{"title":"Several Congruences Related to Harmonic Numbers","authors":"","doi":"10.52280/pujm.2021.530801","DOIUrl":"https://doi.org/10.52280/pujm.2021.530801","url":null,"abstract":"Let p be a prime greater than or equal to 5. In this paper, by using the harmonic numbers and Fermat quotient we establish congruences\u0000involving the sums\u0000p−1 X2\u0000k=1\u0000µ\u0000k\u0000r\u0000¶\u0000Hk,\u0000p−1 X2\u0000k=1\u0000¡\u00002k\u0000k\u0000¢2\u000016k H\u0000(2)\u0000k\u0000and\u0000p−1 X2\u0000k=1\u00001\u00004\u0000k\u0000µ\u00002k\u0000k\u0000¶\u0000H\u0000(3)\u0000k\u0000.\u0000For example,\u0000p−1 X2\u0000k=0\u0000¡\u00002k\u0000k\u0000¢2\u000016k H\u0000(2)\u0000k ≡ 4E2p−4 − 8Ep−3\u0000¡\u0000mod p\u00002\u0000¢\u0000,\u0000where H\u0000(m)\u0000k\u0000are the generalized harmonic numbers of order m and En are\u0000Euler numbers","PeriodicalId":205373,"journal":{"name":"Punjab University Journal of Mathematics","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116580598","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 : 2021-08-26DOI: 10.52280/pujm.2021.530804
: In this paper, we produce a novel framework of a subclass of convex functions that is exponentially convex functions. Moreover, it is observed that the new concept helps to build new inequalities of Petrovic’s ´ type by employing exponentially convex functions. We also introduce the idea of coordinated exponentially convex functions and derive Petrovic’s ´ type inequality for coordinated exponentially convex functions. We also find Lagrange type and Cauchy type mean value theorems for Petrovic’s ´ type inequality for exponentially convex and coordinated exponentially convex functions. Our consequences with this new generalizations has the abilities to be implemented for the evaluation of many mathematical problems.
{"title":"Petrovic’s ´ type inequality for exponentially convex functions and coordinated\u0000exponentially convex functions","authors":"","doi":"10.52280/pujm.2021.530804","DOIUrl":"https://doi.org/10.52280/pujm.2021.530804","url":null,"abstract":": In this paper, we produce a novel framework of a subclass of\u0000convex functions that is exponentially convex functions. Moreover, it is\u0000observed that the new concept helps to build new inequalities of Petrovic’s ´\u0000type by employing exponentially convex functions. We also introduce the\u0000idea of coordinated exponentially convex functions and derive Petrovic’s ´\u0000type inequality for coordinated exponentially convex functions. We also\u0000find Lagrange type and Cauchy type mean value theorems for Petrovic’s ´\u0000type inequality for exponentially convex and coordinated exponentially\u0000convex functions. Our consequences with this new generalizations has\u0000the abilities to be implemented for the evaluation of many mathematical\u0000problems.","PeriodicalId":205373,"journal":{"name":"Punjab University Journal of Mathematics","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116809047","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 : 2021-08-26DOI: 10.52280/pujm.2021.530803
F. Khan, M. Sultana, M. Khalid
The aim of this research was to relate two physical effects for partial differential equations on the time-coordinate, notably the multipledelay times and fractional-derivative. Time Fractional Delay Partial Differential Equations (TFDPDEs) usually interpret some complex physical phenomenon. This study works to solve TFDPDE with shrinking in x and proportional delays in t numerically by utilizing the fractional derivative of Caputo sense in the numerical method known as Perturbation Iteration Algorithm (PIA). A few famous numerical examples have been solved using PIA and their comparison with an exact solutions is illustrated for ® = 1. Also, different values of ® have been depicted in graphical form to show their fractional behavior. The delay term k is also discussed extensively in this TFDPDE study. Numerical results show that this technique is reliable, convenient, and attractive for computational use in modern times.
{"title":"Numerical Solution of Time Fractional Delay Partial Differential Equations\u0000by Perturbation Iteration Algorithm","authors":"F. Khan, M. Sultana, M. Khalid","doi":"10.52280/pujm.2021.530803","DOIUrl":"https://doi.org/10.52280/pujm.2021.530803","url":null,"abstract":"The aim of this research was to relate two physical effects for\u0000partial differential equations on the time-coordinate, notably the multipledelay\u0000times and fractional-derivative. Time Fractional Delay Partial Differential\u0000Equations (TFDPDEs) usually interpret some complex physical\u0000phenomenon. This study works to solve TFDPDE with shrinking in x and\u0000proportional delays in t numerically by utilizing the fractional derivative\u0000of Caputo sense in the numerical method known as Perturbation Iteration\u0000Algorithm (PIA). A few famous numerical examples have been solved\u0000using PIA and their comparison with an exact solutions is illustrated for\u0000® = 1. Also, different values of ® have been depicted in graphical form to\u0000show their fractional behavior. The delay term k is also discussed extensively\u0000in this TFDPDE study. Numerical results show that this technique is\u0000reliable, convenient, and attractive for computational use in modern times.","PeriodicalId":205373,"journal":{"name":"Punjab University Journal of Mathematics","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124838985","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 : 2021-08-25DOI: 10.52280/pujm.2021.530901
The α-Sumudu transform is defined and its properties are proved. α-Sumudu transform of convolution product and composition of functions is obtained. The α-Sumudu transform of Riemann-Liouville integral and derivatives of fractional order are determined. As an application, the solution of Initial Value Problems with Riemann-Liouville derivative of fractional order is obtained. .
{"title":"On Properties of α-Sumudu Transform and Applications","authors":"","doi":"10.52280/pujm.2021.530901","DOIUrl":"https://doi.org/10.52280/pujm.2021.530901","url":null,"abstract":"The α-Sumudu transform is defined and its properties are\u0000proved. α-Sumudu transform of convolution product and composition of\u0000functions is obtained. The α-Sumudu transform of Riemann-Liouville\u0000integral and derivatives of fractional order are determined. As an application, the solution of Initial Value Problems with Riemann-Liouville derivative of fractional order is obtained. .","PeriodicalId":205373,"journal":{"name":"Punjab University Journal of Mathematics","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130000768","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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