This work is focused on the derivation and analysis of a novel numerical technique for solving time fractional mobile-immobile advection-dispersion equation which models many complex systems in engineering and science. The scheme is derived using the effective combination of Euler and Caputo numerical techniques for approximating the integer and fractional time derivatives respectively, and a fourth order exponential compact scheme for spatial derivatives. The Fourier analysis technique is used to prove that the proposed numerical scheme is unconditionally stable and perform convergence analysis. To assess the viability and accuracy of the proposed scheme, some numerical examples are demonstrated with constant as well as variable order time fractional derivatives for this model.
{"title":"A New Compact Numerical Scheme for Solving Time Fractional Mobile-Immobile Advection-Dispersion Model","authors":"S. Thomas, S. K. Nadupuri","doi":"10.47836/mjms.17.3.02","DOIUrl":"https://doi.org/10.47836/mjms.17.3.02","url":null,"abstract":"This work is focused on the derivation and analysis of a novel numerical technique for solving time fractional mobile-immobile advection-dispersion equation which models many complex systems in engineering and science. The scheme is derived using the effective combination of Euler and Caputo numerical techniques for approximating the integer and fractional time derivatives respectively, and a fourth order exponential compact scheme for spatial derivatives. The Fourier analysis technique is used to prove that the proposed numerical scheme is unconditionally stable and perform convergence analysis. To assess the viability and accuracy of the proposed scheme, some numerical examples are demonstrated with constant as well as variable order time fractional derivatives for this model.","PeriodicalId":43645,"journal":{"name":"Malaysian Journal of Mathematical Sciences","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135785547","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}
For a filter T of an MS -algebra L and a subset Z of L, a new extension filter of T is introduced, denoted by ET(Z). Many properties of ET(Z) are investigated and the lattice structure of the set of all ET(Z) is studied. A new definition related to ET(Z) is presented, called fixed filters relative to a subset of L. A generalisation of ET(Z) is illustrated by introducing the concept of strong filters, notated by ET(Z)¯¯¯¯¯¯¯¯¯¯¯¯¯¯. The strong extension ET(Z)¯¯¯¯¯¯¯¯¯¯¯¯¯¯ is characterized by the intersection of all strong filters fixed relative to an ideal L−P for a prime filter P of L.
{"title":"Extended Filters of MS -Algebras","authors":"A. Gaber, M. A. Seoud, M. Tarek","doi":"10.47836/mjms.17.3.13","DOIUrl":"https://doi.org/10.47836/mjms.17.3.13","url":null,"abstract":"For a filter T of an MS -algebra L and a subset Z of L, a new extension filter of T is introduced, denoted by ET(Z). Many properties of ET(Z) are investigated and the lattice structure of the set of all ET(Z) is studied. A new definition related to ET(Z) is presented, called fixed filters relative to a subset of L. A generalisation of ET(Z) is illustrated by introducing the concept of strong filters, notated by ET(Z)¯¯¯¯¯¯¯¯¯¯¯¯¯¯. The strong extension ET(Z)¯¯¯¯¯¯¯¯¯¯¯¯¯¯ is characterized by the intersection of all strong filters fixed relative to an ideal L−P for a prime filter P of L.","PeriodicalId":43645,"journal":{"name":"Malaysian Journal of Mathematical Sciences","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135785548","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}
By applying the standard fractional integral operator of Riemann-Liouville on MT-convex stochastic processes, we can obtain new inequalities of Hermite-Hadamard, providing in the process new estimates on these types of Hermite-Hadamard inequalities for stochastic process whose first derivatives absolute values are MT-convex.
{"title":"Hermite-Hadamard Inequalities Type Using Fractional Integrals for MT-convex Stochastic Process","authors":"O. Rholam, M. Barmaki, D. Gretete","doi":"10.47836/mjms.17.3.14","DOIUrl":"https://doi.org/10.47836/mjms.17.3.14","url":null,"abstract":"By applying the standard fractional integral operator of Riemann-Liouville on MT-convex stochastic processes, we can obtain new inequalities of Hermite-Hadamard, providing in the process new estimates on these types of Hermite-Hadamard inequalities for stochastic process whose first derivatives absolute values are MT-convex.","PeriodicalId":43645,"journal":{"name":"Malaysian Journal of Mathematical Sciences","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135785549","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 aim of this manuscript is to solve the initial-value problems of neutral delay Volterra integro-differential equations with constant or proportional delays. Hence, a proposed hybrid technique named as an implicit multistep block method with an off-step point (1OBM4) is formulated for the numerical solution of NDVIDE. A LMM associated with an off-point is known as hybrid LMM. The proposed technique, 1OBM4, attempts to solve the problem synchronously in a block manner. Moreover, a Taylor expansion is implemented to develop 1OBM4 in predictor-corrector mode. Two different approaches are presented in order to solve both integral and differential parts of the problem. Some analyses on 1OBM4 are considered in terms of order and convergence of the method. A stability polynomial is also obtained for the stability regions to be constructed. In the last section, some numerical results are demonstrated to show the applicability of 1OBM4 in solving NDVIDE with constant or proportional delays.
{"title":"The Implicit Multistep Block Method with An Off-step Point for Initial Value Problems of Neutral Delay Volterra Integro-differential Equations","authors":"N. I. N. Ismail, Z. A. Majid","doi":"10.47836/mjms.17.3.15","DOIUrl":"https://doi.org/10.47836/mjms.17.3.15","url":null,"abstract":"The aim of this manuscript is to solve the initial-value problems of neutral delay Volterra integro-differential equations with constant or proportional delays. Hence, a proposed hybrid technique named as an implicit multistep block method with an off-step point (1OBM4) is formulated for the numerical solution of NDVIDE. A LMM associated with an off-point is known as hybrid LMM. The proposed technique, 1OBM4, attempts to solve the problem synchronously in a block manner. Moreover, a Taylor expansion is implemented to develop 1OBM4 in predictor-corrector mode. Two different approaches are presented in order to solve both integral and differential parts of the problem. Some analyses on 1OBM4 are considered in terms of order and convergence of the method. A stability polynomial is also obtained for the stability regions to be constructed. In the last section, some numerical results are demonstrated to show the applicability of 1OBM4 in solving NDVIDE with constant or proportional delays.","PeriodicalId":43645,"journal":{"name":"Malaysian Journal of Mathematical Sciences","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135785551","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}
N. T. Jaaffar, N. I. N. Ismail, Z. A. Majid, N. Senu
The boundary and initial conditions that are related to the retarded and neutral delay differential equations, respectively, will be resolved in this work by using the previous direct multistep method. This method solves retarded and neutral delay differential equations directly by implementing the proposed method without converting it to a first-order system. For boundary value problems, the shooting strategy incorporated with the Newton method is utilized to predict the guessing value. The initial value problem for neutral delay differential equations on the other hand is resolved directly with special attention to the differential part of the problem. Several numerical examples are investigated to observe the capability of the developed strategies and methods for solving retarded delay differential equations with boundary value problems and neutral delay differential equations with initial value problems.
{"title":"Direct Multistep Method for Solving Retarded and Neutral Delay Differential Equation with Boundary and Initial Value Problems","authors":"N. T. Jaaffar, N. I. N. Ismail, Z. A. Majid, N. Senu","doi":"10.47836/mjms.17.3.10","DOIUrl":"https://doi.org/10.47836/mjms.17.3.10","url":null,"abstract":"The boundary and initial conditions that are related to the retarded and neutral delay differential equations, respectively, will be resolved in this work by using the previous direct multistep method. This method solves retarded and neutral delay differential equations directly by implementing the proposed method without converting it to a first-order system. For boundary value problems, the shooting strategy incorporated with the Newton method is utilized to predict the guessing value. The initial value problem for neutral delay differential equations on the other hand is resolved directly with special attention to the differential part of the problem. Several numerical examples are investigated to observe the capability of the developed strategies and methods for solving retarded delay differential equations with boundary value problems and neutral delay differential equations with initial value problems.","PeriodicalId":43645,"journal":{"name":"Malaysian Journal of Mathematical Sciences","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135785544","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 at solving the optimal control problem of the dynamic of HBV infection under treatment using the perturbation iteration method. This method serves as a tool to determine the approximate solutions of nonlinear equations for which exact solutions cannot be obtained. To test the efficacy of this method, the authors propose to compare the numerical simulation results with those of the direct method and fuzzy logic strategy. The newly used method for solving the above optimal control problem is very important since the findings compared to those obtained from the two other methods are in good agreement with experimental data and they demonstrate the response drugs to the dynamics of uninfected hepatocytes, infected hepatocytes, and free virions for a patient suffering from HBV. Since the perturbation iteration method provides satisfactory results which are close to other used numerical methods, it is an important numerical tool to determine the solution of an optimal control problem. In particular, it provides optimal trajectories in medicine, biology, and other related scientific fields. For instance, the response of treatment as control of the human body ensures the health of patients.
{"title":"Perturbation Iteration Method Compared with Direct Method and Fuzzy Logic Strategy for Solving An Optimal Control Problem of An Uninfected Hepatitis B Virus Dynamics","authors":"Daoussa Haggar Mahamat Saleh, Jean Marie Ntaganda","doi":"10.47836/mjms.17.3.01","DOIUrl":"https://doi.org/10.47836/mjms.17.3.01","url":null,"abstract":"This paper aims at solving the optimal control problem of the dynamic of HBV infection under treatment using the perturbation iteration method. This method serves as a tool to determine the approximate solutions of nonlinear equations for which exact solutions cannot be obtained. To test the efficacy of this method, the authors propose to compare the numerical simulation results with those of the direct method and fuzzy logic strategy. The newly used method for solving the above optimal control problem is very important since the findings compared to those obtained from the two other methods are in good agreement with experimental data and they demonstrate the response drugs to the dynamics of uninfected hepatocytes, infected hepatocytes, and free virions for a patient suffering from HBV. Since the perturbation iteration method provides satisfactory results which are close to other used numerical methods, it is an important numerical tool to determine the solution of an optimal control problem. In particular, it provides optimal trajectories in medicine, biology, and other related scientific fields. For instance, the response of treatment as control of the human body ensures the health of patients.","PeriodicalId":43645,"journal":{"name":"Malaysian Journal of Mathematical Sciences","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135785704","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 purpose of this study is to introduce the wave structures and dynamical features of the third-order nonlinear Schr"{o}dinger equations (TONLSE). We take the original equation and, using the traveling wave transformation, convert it into the appropriate traveling wave system, from which we create a conserved quantity known as the Hamiltonian. The Jacobian elliptic function solution (JEF), the hyperbolic function solution, and the trigonometric function solution are just a few of the optical soliton solutions to the equation that may be found using the complete discrimination system (CDS) of polynomial method (CDSPM) and also transfer the JEF into solitary wave (SW) soltions. It also includes certain dynamic results, such as bifurcation points and critical conditions for solutions, that might be utilized to explore the dynamic features of the equation employing the CDSPM. This method could also be used for qualitative analysis. The qualitative analysis is used to illustrate the equilibrium points and phase potraits of the equation. Phase portraits are visual representations used in dynamical systems to illustrate a system's behaviour through time. They can provide crucial information about a system's stability, periodic behaviour, and the presence of attractors or repellents.
{"title":"Dynamical Discussion and Diverse Soliton Solutions via Complete Discrimination System Approach Along with Bifurcation Analysis for the Third Order NLSE","authors":"S. T. R. Rizvi, A. R. Seadawy, B. Mustafa","doi":"10.47836/mjms.17.3.09","DOIUrl":"https://doi.org/10.47836/mjms.17.3.09","url":null,"abstract":"The purpose of this study is to introduce the wave structures and dynamical features of the third-order nonlinear Schr\"{o}dinger equations (TONLSE). We take the original equation and, using the traveling wave transformation, convert it into the appropriate traveling wave system, from which we create a conserved quantity known as the Hamiltonian. The Jacobian elliptic function solution (JEF), the hyperbolic function solution, and the trigonometric function solution are just a few of the optical soliton solutions to the equation that may be found using the complete discrimination system (CDS) of polynomial method (CDSPM) and also transfer the JEF into solitary wave (SW) soltions. It also includes certain dynamic results, such as bifurcation points and critical conditions for solutions, that might be utilized to explore the dynamic features of the equation employing the CDSPM. This method could also be used for qualitative analysis. The qualitative analysis is used to illustrate the equilibrium points and phase potraits of the equation. Phase portraits are visual representations used in dynamical systems to illustrate a system's behaviour through time. They can provide crucial information about a system's stability, periodic behaviour, and the presence of attractors or repellents.","PeriodicalId":43645,"journal":{"name":"Malaysian Journal of Mathematical Sciences","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135785552","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}
Z. Bouazza, M. S. Souid, C. H. C. Hussin, A. Mandangan, S. Sabit
In this manuscript, we examine the existence and the stability of solutions to the boundary value problem of Riemann-Liouville fractional differential equations of variable order. The obtained new results are based on the fixed point theorem of Darbo and Kuratowski’s metric of noncompactness (MNK) with the help of piece-wise constant functions. In addition, the derived fundamental results are proven suitable because they satisfy the Ulam-Hyers Rassias stability sufficient conditions. Several numerical examples were discussed too to demonstrate the reasonableness and effectiveness of the observed results.
{"title":"Variable-order Implicit Fractional Differential Equations based on the Kuratowski MNC Technique","authors":"Z. Bouazza, M. S. Souid, C. H. C. Hussin, A. Mandangan, S. Sabit","doi":"10.47836/mjms.17.3.05","DOIUrl":"https://doi.org/10.47836/mjms.17.3.05","url":null,"abstract":"In this manuscript, we examine the existence and the stability of solutions to the boundary value problem of Riemann-Liouville fractional differential equations of variable order. The obtained new results are based on the fixed point theorem of Darbo and Kuratowski’s metric of noncompactness (MNK) with the help of piece-wise constant functions. In addition, the derived fundamental results are proven suitable because they satisfy the Ulam-Hyers Rassias stability sufficient conditions. Several numerical examples were discussed too to demonstrate the reasonableness and effectiveness of the observed results.","PeriodicalId":43645,"journal":{"name":"Malaysian Journal of Mathematical Sciences","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135785701","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}
Analogical reasoning plays a crucial part in problem-solving since it requires students to connect prior knowledge with the issues at hand in learning mathematics. However, students struggle when developing solutions to the issues utilizing analogies even if there is a connection between mathematical creativity and analogical reasoning. The aims of this study were to assess students' use of Ruppert's phases to solve problems and identify students' analogy patterns to solve target problems. This study is qualitative in nature. Of 19 research participants, six were then chosen using the purposive sampling technique based on their levels of mathematical creative ability. Test, interview, and documentation were the data gathering techniques used in this study. The study's findings suggested that good analogical reasoning skills did not serve as a prerequisite for students with strong mathematical creative thinking skills. Only one subject out of three who possessed necessary mathematical creative thinking abilities could go through the four steps of analogical reasoning-structuring, mapping, applying, and verifying. All other subjects were unable to complete the four steps of analogy, and even their creative thinking skills were weak. This was because the students did not comprehend the idea and could not connect prior knowledge with the issues at hand. In order to remind students of their prior knowledge and experiences, it would therefore be necessary at this analogy stage to establish an initial stage before structuring. The format and degree of difficulty of the questions were assumed to be other elements that might influence students' responses. The results of this study are expected to be a reference for further research, namely increasing analogical reasoning optimally as an effort to increase students' prior knowledge and students' mathematical creative thinking abilities in solving mathematical problems.
{"title":"Students' Analogical Reasoning in Solving Trigonometric Target Problems","authors":"None Mutia, None Kartono, None Dwijanto, None Kristina Wijayanti","doi":"10.47836/mjms.17.3.11","DOIUrl":"https://doi.org/10.47836/mjms.17.3.11","url":null,"abstract":"Analogical reasoning plays a crucial part in problem-solving since it requires students to connect prior knowledge with the issues at hand in learning mathematics. However, students struggle when developing solutions to the issues utilizing analogies even if there is a connection between mathematical creativity and analogical reasoning. The aims of this study were to assess students' use of Ruppert's phases to solve problems and identify students' analogy patterns to solve target problems. This study is qualitative in nature. Of 19 research participants, six were then chosen using the purposive sampling technique based on their levels of mathematical creative ability. Test, interview, and documentation were the data gathering techniques used in this study. The study's findings suggested that good analogical reasoning skills did not serve as a prerequisite for students with strong mathematical creative thinking skills. Only one subject out of three who possessed necessary mathematical creative thinking abilities could go through the four steps of analogical reasoning-structuring, mapping, applying, and verifying. All other subjects were unable to complete the four steps of analogy, and even their creative thinking skills were weak. This was because the students did not comprehend the idea and could not connect prior knowledge with the issues at hand. In order to remind students of their prior knowledge and experiences, it would therefore be necessary at this analogy stage to establish an initial stage before structuring. The format and degree of difficulty of the questions were assumed to be other elements that might influence students' responses. The results of this study are expected to be a reference for further research, namely increasing analogical reasoning optimally as an effort to increase students' prior knowledge and students' mathematical creative thinking abilities in solving mathematical problems.","PeriodicalId":43645,"journal":{"name":"Malaysian Journal of Mathematical Sciences","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135785550","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 state and prove a condition for the local stability of a certain class of two dimensional system of polynomial differential equations. We give some examples of polynomial differential systems of equations to demonstrate that this local stability condition established for the trivial equilibrium point (0,0) is quite sharp and compare our result with the well known Lyapunov local stability criterion (Lyapunov's second method).
{"title":"On the Local Stability of a Certain Class of Polynomial Differential System","authors":"Ismail Mirumbe, J. Nakakawa, J. Mango","doi":"10.47836/mjms.17.2.06","DOIUrl":"https://doi.org/10.47836/mjms.17.2.06","url":null,"abstract":"We state and prove a condition for the local stability of a certain class of two dimensional system of polynomial differential equations. We give some examples of polynomial differential systems of equations to demonstrate that this local stability condition established for the trivial equilibrium point (0,0) is quite sharp and compare our result with the well known Lyapunov local stability criterion (Lyapunov's second method).","PeriodicalId":43645,"journal":{"name":"Malaysian Journal of Mathematical Sciences","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44504779","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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