In this article, we introduce a novel spectral algorithm utilizing Fibonacci polynomials to numerically solve both linear and nonlinear integro-differential equations with fractional-order derivatives. Our approach employs a quadrature-collocation method, transforming complex equations and associated conditions into systems of linear or nonlinear algebraic equations. The solutions to these equations, involving unknown coefficients, provide accurate numerical approximations for the original fractional-order equations. To validate the method, we present numerical examples illustrating its robustness and versatility. Comparative analyses with available analytical solutions affirm the reliability and accuracy of our algorithm, establishing its practical utility in addressing fractional-order integro-differential equations. This research contributes to computational mathematics and spectral methods, offering a promising tool for diverse scientific and engineering challenges.
{"title":"Fejér-Quadrature Collocation Algorithm for Solving Fractional Integro-Differential Equations via Fibonacci Polynomials","authors":"Y. H. Youssri, A. G. Atta","doi":"10.37256/cm.5120244054","DOIUrl":"https://doi.org/10.37256/cm.5120244054","url":null,"abstract":"In this article, we introduce a novel spectral algorithm utilizing Fibonacci polynomials to numerically solve both linear and nonlinear integro-differential equations with fractional-order derivatives. Our approach employs a quadrature-collocation method, transforming complex equations and associated conditions into systems of linear or nonlinear algebraic equations. The solutions to these equations, involving unknown coefficients, provide accurate numerical approximations for the original fractional-order equations. To validate the method, we present numerical examples illustrating its robustness and versatility. Comparative analyses with available analytical solutions affirm the reliability and accuracy of our algorithm, establishing its practical utility in addressing fractional-order integro-differential equations. This research contributes to computational mathematics and spectral methods, offering a promising tool for diverse scientific and engineering challenges.","PeriodicalId":504505,"journal":{"name":"Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139618972","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}
Vaccination programs aimed at preventing the spread of the coronavirus appear to have a significant global impact. In this research, we have investigated a mathematical model projecting COVID-19 disease spread by considering five groups of individuals viz. vulnerable, exposed, infected, unreported, recovered, and vaccinated. Looking at the current abnormal pattern of the virus spread in the projected model, we have implemented the fractional derivative in the Mittag-Leffler context. Using the existing theory of the fractional derivative, we have examined the theoretical aspects such as the existence and uniqueness of the solutions, the existence and stability of the disease-free and endemic equilibrium points, and the global stability of the disease-free equilibrium point. In computing the basic reproduction number, we have analyzed that the existence and stability of points of equilibrium are dependent on this number. The sensitivity of the basic reproduction number is also examined. The importance of the vaccination drive is highlighted by relating it to the basic reproduction number. Finally, we have presented the simulation of the numerical results by capturing the profile of each group under the influence of the fractional derivative and investigated the impact of vaccination rate and contact rate in controlling the disease by applying the Adams-Bashforth-Moultan (ABM) method. The present research study demonstrates the importance of the vaccination campaign and the curb on individual contact by featuring a novel fractional operator in the projected model and capturing the corresponding consequence.
{"title":"An Application of the Caputo Fractional Domain in the Analysis of a COVID-19 Mathematical Model","authors":"Chandrali Baishya, Sindhu J. Achar, P. Veeresha","doi":"10.37256/cm.5120242363","DOIUrl":"https://doi.org/10.37256/cm.5120242363","url":null,"abstract":"Vaccination programs aimed at preventing the spread of the coronavirus appear to have a significant global impact. In this research, we have investigated a mathematical model projecting COVID-19 disease spread by considering five groups of individuals viz. vulnerable, exposed, infected, unreported, recovered, and vaccinated. Looking at the current abnormal pattern of the virus spread in the projected model, we have implemented the fractional derivative in the Mittag-Leffler context. Using the existing theory of the fractional derivative, we have examined the theoretical aspects such as the existence and uniqueness of the solutions, the existence and stability of the disease-free and endemic equilibrium points, and the global stability of the disease-free equilibrium point. In computing the basic reproduction number, we have analyzed that the existence and stability of points of equilibrium are dependent on this number. The sensitivity of the basic reproduction number is also examined. The importance of the vaccination drive is highlighted by relating it to the basic reproduction number. Finally, we have presented the simulation of the numerical results by capturing the profile of each group under the influence of the fractional derivative and investigated the impact of vaccination rate and contact rate in controlling the disease by applying the Adams-Bashforth-Moultan (ABM) method. The present research study demonstrates the importance of the vaccination campaign and the curb on individual contact by featuring a novel fractional operator in the projected model and capturing the corresponding consequence.","PeriodicalId":504505,"journal":{"name":"Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139437717","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}
A. Manickam, M. Kavitha, A. Benevatho Jaison, Arvind Kumar Singh
This article investigates a fractional-order mathematical model of Banana Xanthomonas Wilt disease while considering control measures using Caputo derivatives. The proposed model is numerically solved using the L1-based predictor-corrector method to explore the model’s dynamics in a particular time range. Stability and error analyses are performed to justify the efficiency of the scheme. The non-local nature of the Caputo fractional derivative, which includes memory effects in the system, is the main motivation for incorporating this derivative in the model. We obtain varieties in the model dynamics while checking various fractional order values.
{"title":"A Fractional-Order Mathematical Model of Banana Xanthomonas Wilt Disease Using Caputo Derivatives","authors":"A. Manickam, M. Kavitha, A. Benevatho Jaison, Arvind Kumar Singh","doi":"10.37256/cm.5120242479","DOIUrl":"https://doi.org/10.37256/cm.5120242479","url":null,"abstract":"This article investigates a fractional-order mathematical model of Banana Xanthomonas Wilt disease while considering control measures using Caputo derivatives. The proposed model is numerically solved using the L1-based predictor-corrector method to explore the model’s dynamics in a particular time range. Stability and error analyses are performed to justify the efficiency of the scheme. The non-local nature of the Caputo fractional derivative, which includes memory effects in the system, is the main motivation for incorporating this derivative in the model. We obtain varieties in the model dynamics while checking various fractional order values.","PeriodicalId":504505,"journal":{"name":"Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139442435","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 objective of this work is to solve a coronavirus transmission model using simulink, a platform for model based design that facilitates simulation and design at the system level. The simulation is divided into two sections: the first, deals with setting up the parameters, and the second deals with computing the fractional derivatives. The fundamental SIR, SEIR, SEIQR, and SEIARM models were used in this study. An effective, quick, easy, and visually appealing method is used to simulate pandemic outbreaks. To accurately follow the progress of the infection, we contrast the simulation findings with those acquired by MATLAB code. The applications can be used for research projects and also as a teaching tool.
{"title":"Simulink Methods to Simulate COVID-19 Outbreak Using Fractional Order Models","authors":"Srilekha R, Parthiban V","doi":"10.37256/cm.5120242591","DOIUrl":"https://doi.org/10.37256/cm.5120242591","url":null,"abstract":"The objective of this work is to solve a coronavirus transmission model using simulink, a platform for model based design that facilitates simulation and design at the system level. The simulation is divided into two sections: the first, deals with setting up the parameters, and the second deals with computing the fractional derivatives. The fundamental SIR, SEIR, SEIQR, and SEIARM models were used in this study. An effective, quick, easy, and visually appealing method is used to simulate pandemic outbreaks. To accurately follow the progress of the infection, we contrast the simulation findings with those acquired by MATLAB code. The applications can be used for research projects and also as a teaching tool.","PeriodicalId":504505,"journal":{"name":"Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139444119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, a novel higher order iterative method for solving nonlinear equations is developed. The new iterative method obtained from fifth order Newton-Özban method attains eighth order of convergence by adding a single step with only one additional function evaluation. The method is extended to Banach spaces and its local as well as semi-local convergence analysis is done under generalized continuity conditions. The existence and uniqueness results of solution are also provided along with radii of convergence balls. From the numerical experiments, it can be inferred that the proposed method is more accurate and effective in high precision computations than existing eighth order methods. The computation of error analysis and norm of functions demonstrate that proposed method takes a lead over the considered methods.
{"title":"Extended Higher Order Iterative Method for Nonlinear Equations and its Convergence Analysis in Banach Spaces","authors":"Gagan Deep, I. Argyros, Gaurav Verma, Simardeep Kaur, Rajdeep Kaur, Samundra Regmi","doi":"10.37256/cm.5120243866","DOIUrl":"https://doi.org/10.37256/cm.5120243866","url":null,"abstract":"In this article, a novel higher order iterative method for solving nonlinear equations is developed. The new iterative method obtained from fifth order Newton-Özban method attains eighth order of convergence by adding a single step with only one additional function evaluation. The method is extended to Banach spaces and its local as well as semi-local convergence analysis is done under generalized continuity conditions. The existence and uniqueness results of solution are also provided along with radii of convergence balls. From the numerical experiments, it can be inferred that the proposed method is more accurate and effective in high precision computations than existing eighth order methods. The computation of error analysis and norm of functions demonstrate that proposed method takes a lead over the considered methods.","PeriodicalId":504505,"journal":{"name":"Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139447746","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}
Convertible bonds are popular financial instruments by which firms raise capital. Owing to the various features of such bonds, especially the early-exercise call, put, and conversion provisions, they can be valued by numerical techniques only. The price of a convertible bond is driven by both the underlying stock price and the interest rate, and these two factors are correlated. Under the partial differential equation framework, a two-dimensional convection-diffusion-reaction equation containing a mixed derivative must be solved. In this work, we employ an Alternating-Direction-Implicit method, namely the Craig-Sneyd scheme to solve the two-factor pricing equation. Comparison against the commonly employed Crank-Nicolson method shows the merit of the scheme. Besides, we analyze how the different contractual features of a convertible bond affect its price.
{"title":"Numerical PDE-Based Pricing of Convertible Bonds Under Two-Factor Models","authors":"R. Coonjobeharry, D. Behera, N. Thakoor","doi":"10.37256/cm.5120243343","DOIUrl":"https://doi.org/10.37256/cm.5120243343","url":null,"abstract":"Convertible bonds are popular financial instruments by which firms raise capital. Owing to the various features of such bonds, especially the early-exercise call, put, and conversion provisions, they can be valued by numerical techniques only. The price of a convertible bond is driven by both the underlying stock price and the interest rate, and these two factors are correlated. Under the partial differential equation framework, a two-dimensional convection-diffusion-reaction equation containing a mixed derivative must be solved. In this work, we employ an Alternating-Direction-Implicit method, namely the Craig-Sneyd scheme to solve the two-factor pricing equation. Comparison against the commonly employed Crank-Nicolson method shows the merit of the scheme. Besides, we analyze how the different contractual features of a convertible bond affect its price.","PeriodicalId":504505,"journal":{"name":"Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139453395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we introduce a novel, all-purpose entropy measure and derive a new, all-purpose average codeword length in accordance with it. After that, we determined new average generalized codeword length constraints in terms of additional metrics of generalized entropy. In addition, we demonstrate that the metrics employed in this study are generalizations of other metrics typically employed in the coding and information theory communities.
{"title":"Novel Generalized Entropy Measure's Characteristics Associated with Code-Word Length","authors":"Aakanksha Dwivedi, R. N. Saraswat","doi":"10.37256/cm.5120243281","DOIUrl":"https://doi.org/10.37256/cm.5120243281","url":null,"abstract":"In this article, we introduce a novel, all-purpose entropy measure and derive a new, all-purpose average codeword length in accordance with it. After that, we determined new average generalized codeword length constraints in terms of additional metrics of generalized entropy. In addition, we demonstrate that the metrics employed in this study are generalizations of other metrics typically employed in the coding and information theory communities.","PeriodicalId":504505,"journal":{"name":"Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139452644","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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