In this study, the Lie symmetry analysis is given for the time-fractional telegraph equation with the Riemann–Liouville derivative. This equation is useable to describe the physical processes of models possessing memory. By applying classical and nonclassical Lie symmetry analysis for the telegraph equation with α , β time-fractional derivatives and some technical computations, new infinitesimal generators are obtained. The actual methods give some classical symmetries while the nonclassical approach will bring back other symmetries to these equations. The similarity reduction and conservation laws to the fractional telegraph equation are found.
{"title":"Invariant Solutions and Conservation Laws of the Time-Fractional Telegraph Equation","authors":"R. Najafi, E. Çelik, Neslihan Uyanik","doi":"10.1155/2023/1294070","DOIUrl":"https://doi.org/10.1155/2023/1294070","url":null,"abstract":"In this study, the Lie symmetry analysis is given for the time-fractional telegraph equation with the Riemann–Liouville derivative. This equation is useable to describe the physical processes of models possessing memory. By applying classical and nonclassical Lie symmetry analysis for the telegraph equation with \u0000 \u0000 α\u0000 ,\u0000 β\u0000 \u0000 time-fractional derivatives and some technical computations, new infinitesimal generators are obtained. The actual methods give some classical symmetries while the nonclassical approach will bring back other symmetries to these equations. The similarity reduction and conservation laws to the fractional telegraph equation are found.","PeriodicalId":49111,"journal":{"name":"Advances in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48492360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper uses Müntz orthogonal functions to numerically solve the fractional Bagley–Torvik equation with initial and boundary conditions. Müntz orthogonal functions are defined on the interval 0 , 1 and have simple and distinct real roots on this interval. For the function f ∈ L 2 0 , 1 , we obtain the best unique approximation using Müntz orthogonal functions. We obtain the Riemann–Liouville fractional integral operator for Müntz orthogonal functions so that we can reduce the complexity of calculations and increase the speed of solving the problem, which can be seen in the process of running the Maple program. To solve the fractional Bagley–Torvik equation with initial and boundary conditions, we use Müntz orthogonal functions and consider simple and distinct real roots of Müntz orthogonal functions as collocation points. By using the Riemann–Liouville fractional integral operator that we define for the Müntz orthogonal functions, the process of numerically solving the fractional Bagley–Torvik equation that is solved using Müntz orthogonal functions is reduced, and finally, we reach a system of algebraic equations. By solving algebraic equations and obtaining the vector of unknowns, the fractional Bagley–Torvik equation is solved using Müntz orthogonal functions, and the error value of the method can be calculated. The low error value of this numerical solution method shows the high accuracy of this method. With the help of the Müntz functions, we obtain the error bound for the approximation of the function. We have obtained the error bounds for the numerical method using which we solved the fractional Bagley–Torvik equation with initial and boundary conditions. Finally, we have given a numerical example to show the accuracy of the solution of the method presented in this paper. The results of solving this example using Müntz orthogonal functions and comparing the results with other methods that have been used the solve this example show the higher accuracy of the method proposed in this paper.
{"title":"Application of Müntz Orthogonal Functions on the Solution of the Fractional Bagley–Torvik Equation Using Collocation Method with Error Stimate","authors":"S. Akhlaghi, M. Tavassoli Kajani, M. Allame","doi":"10.1155/2023/5520787","DOIUrl":"https://doi.org/10.1155/2023/5520787","url":null,"abstract":"This paper uses Müntz orthogonal functions to numerically solve the fractional Bagley–Torvik equation with initial and boundary conditions. Müntz orthogonal functions are defined on the interval \u0000 \u0000 \u0000 \u0000 0\u0000 ,\u0000 1\u0000 \u0000 \u0000 \u0000 and have simple and distinct real roots on this interval. For the function \u0000 \u0000 f\u0000 ∈\u0000 \u0000 L\u0000 \u0000 2\u0000 \u0000 \u0000 \u0000 \u0000 0\u0000 ,\u0000 1\u0000 \u0000 \u0000 \u0000 , we obtain the best unique approximation using Müntz orthogonal functions. We obtain the Riemann–Liouville fractional integral operator for Müntz orthogonal functions so that we can reduce the complexity of calculations and increase the speed of solving the problem, which can be seen in the process of running the Maple program. To solve the fractional Bagley–Torvik equation with initial and boundary conditions, we use Müntz orthogonal functions and consider simple and distinct real roots of Müntz orthogonal functions as collocation points. By using the Riemann–Liouville fractional integral operator that we define for the Müntz orthogonal functions, the process of numerically solving the fractional Bagley–Torvik equation that is solved using Müntz orthogonal functions is reduced, and finally, we reach a system of algebraic equations. By solving algebraic equations and obtaining the vector of unknowns, the fractional Bagley–Torvik equation is solved using Müntz orthogonal functions, and the error value of the method can be calculated. The low error value of this numerical solution method shows the high accuracy of this method. With the help of the Müntz functions, we obtain the error bound for the approximation of the function. We have obtained the error bounds for the numerical method using which we solved the fractional Bagley–Torvik equation with initial and boundary conditions. Finally, we have given a numerical example to show the accuracy of the solution of the method presented in this paper. The results of solving this example using Müntz orthogonal functions and comparing the results with other methods that have been used the solve this example show the higher accuracy of the method proposed in this paper.","PeriodicalId":49111,"journal":{"name":"Advances in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43515610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Retracted: Use Python Data Analysis to Gain Insights from Airbnb Hosts","authors":"Advances in Mathematical Physics","doi":"10.1155/2023/9893030","DOIUrl":"https://doi.org/10.1155/2023/9893030","url":null,"abstract":"<jats:p />","PeriodicalId":49111,"journal":{"name":"Advances in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45223338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the longitudinal model of an uncertain aircraft is taken as the research object, and the aircraft path inclination is controlled by controlling the input rudder deflection angle. An adaptive iterative learning control (AILC) scheme is proposed to solve the accurate tracking control problem of the flight path inclination on a finite time interval. The aircraft track angle system is abstractly modeled to obtain a triangular model in the form of strict feedback. For the abstracted strict feedback model, the fuzzy logic is used to approximate the uncertain part of the model. A command filter and an error compensation mechanism are introduced to prevent the computational bloat problem caused by excessive system order, and a convergent series sequence is used to deal with the truncation error caused by the approximation of the fuzzy logic. Based on the Lyapunov stability theorem, all signals of the closed-loop system are bounded on the finite time interval 0 , T , and the output of the system can track the desired trajectory accurately. Finally, the feasibility and effectiveness of the method are verified by MATLAB simulation results.
{"title":"Command Filter AILC for Finite Time Accurate Tracking of Aircraft Track Angle System Based on Fuzzy Logic","authors":"Chunli Zhang, Xu Tian, Yangjie Gao, F. Qian","doi":"10.1155/2023/4744873","DOIUrl":"https://doi.org/10.1155/2023/4744873","url":null,"abstract":"In this paper, the longitudinal model of an uncertain aircraft is taken as the research object, and the aircraft path inclination is controlled by controlling the input rudder deflection angle. An adaptive iterative learning control (AILC) scheme is proposed to solve the accurate tracking control problem of the flight path inclination on a finite time interval. The aircraft track angle system is abstractly modeled to obtain a triangular model in the form of strict feedback. For the abstracted strict feedback model, the fuzzy logic is used to approximate the uncertain part of the model. A command filter and an error compensation mechanism are introduced to prevent the computational bloat problem caused by excessive system order, and a convergent series sequence is used to deal with the truncation error caused by the approximation of the fuzzy logic. Based on the Lyapunov stability theorem, all signals of the closed-loop system are bounded on the finite time interval \u0000 \u0000 \u0000 \u0000 0\u0000 ,\u0000 T\u0000 \u0000 \u0000 \u0000 , and the output of the system can track the desired trajectory accurately. Finally, the feasibility and effectiveness of the method are verified by MATLAB simulation results.","PeriodicalId":49111,"journal":{"name":"Advances in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47142404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
There are some previous works on designing efficient and high-order numerical methods of density estimation for stochastic partial differential equation (SPDE) driven by multivariate Gaussian random variables. They mostly focus on proposing numerical methods of density estimation for SPDE with independent random variables and rarely research density estimation for SPDE is driven by multivariate Gaussian random variables. In this paper, we propose a high-order algorithm of gPC-based density estimation where SPDE driven by multivariate Gaussian random variables. Our main techniques are (1) we build a new multivariate orthogonal basis by adopting the Gauss–Schmidt orthogonalization; (2) with the newly constructed orthogonal basis in hand, we first assume the unknown function in the SPDE has the stochastic general polynomial chaos (gPC) expansion, second implement the stochastic gPC expansion for the SPDE in the multivariate Gaussian measure space, and third we obtain and numerical calculation deterministic differential equations for the coefficients of the expansion; (3) we used high-order algorithm of gPC-based for density estimation and moment estimation. We apply the newly proposed numerical method to a known random function, stochastic 1D wave equation, and stochastic 2D Schnakenberg model, respectively. All the presented stochastic equations are driven by bivariate Gaussian random variables. The efficiency is compared with the Monte-Carlo method based on the known random function.
{"title":"High-Order Spectral Method of Density Estimation for Stochastic Differential Equation Driven by Multivariate Gaussian Random Variables","authors":"Hongling Xie","doi":"10.1155/2023/9974539","DOIUrl":"https://doi.org/10.1155/2023/9974539","url":null,"abstract":"There are some previous works on designing efficient and high-order numerical methods of density estimation for stochastic partial differential equation (SPDE) driven by multivariate Gaussian random variables. They mostly focus on proposing numerical methods of density estimation for SPDE with independent random variables and rarely research density estimation for SPDE is driven by multivariate Gaussian random variables. In this paper, we propose a high-order algorithm of gPC-based density estimation where SPDE driven by multivariate Gaussian random variables. Our main techniques are (1) we build a new multivariate orthogonal basis by adopting the Gauss–Schmidt orthogonalization; (2) with the newly constructed orthogonal basis in hand, we first assume the unknown function in the SPDE has the stochastic general polynomial chaos (gPC) expansion, second implement the stochastic gPC expansion for the SPDE in the multivariate Gaussian measure space, and third we obtain and numerical calculation deterministic differential equations for the coefficients of the expansion; (3) we used high-order algorithm of gPC-based for density estimation and moment estimation. We apply the newly proposed numerical method to a known random function, stochastic 1D wave equation, and stochastic 2D Schnakenberg model, respectively. All the presented stochastic equations are driven by bivariate Gaussian random variables. The efficiency is compared with the Monte-Carlo method based on the known random function.","PeriodicalId":49111,"journal":{"name":"Advances in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44104158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this manuscript, we introduce the notions of fuzzy strong controlled metric spaces, fuzzy strong controlled quasi-metric spaces, and non-Archimedean fuzzy strong controlled quasi-metric spaces and generalize the famous Banach contraction principle. We prove several fixed point results in the context of non-Archimedean fuzzy strong controlled quasi-metric space. Furthermore, we use our main result to obtain the existence of a solution for a recurrence problem linked with the study of Quicksort algorithms.
{"title":"Fixed Point Results in Fuzzy Strong Controlled Metric Spaces with an Application to the Domain Words","authors":"Aftab Hussain, Umar Ishtiaq, Hamed Al Sulami","doi":"10.1155/2023/4350504","DOIUrl":"https://doi.org/10.1155/2023/4350504","url":null,"abstract":"In this manuscript, we introduce the notions of fuzzy strong controlled metric spaces, fuzzy strong controlled quasi-metric spaces, and non-Archimedean fuzzy strong controlled quasi-metric spaces and generalize the famous Banach contraction principle. We prove several fixed point results in the context of non-Archimedean fuzzy strong controlled quasi-metric space. Furthermore, we use our main result to obtain the existence of a solution for a recurrence problem linked with the study of Quicksort algorithms.","PeriodicalId":49111,"journal":{"name":"Advances in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41594584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Neutrosophic logic is frequently applied to the engineering technology, scientific administration, and financial matters, among other fields. In addition, neutrosophic linear systems can be used to illustrate various practical problems. Due to the complexity of neutrosophic operators, however, solving linear neutrosophic systems is challenging. This work proposes a new straightforward method for solving the neutrosophic system of linear equations based on the neutrosophic structured element (NSE). Here unknown and right-hand side vectors are considered as triangular neutrosophic numbers. Based on the NSE, analytical expressions of the solution to this equation and its degrees are also provided. Finally, several examples of the methodology are provided.
{"title":"An Efficient Technique for Algebraic System of Linear Equations Based on Neutrosophic Structured Element","authors":"Wenbo Xu, Qunli Xia, Hitesh Mohapatra, Sangay Chedup","doi":"10.1155/2023/4469908","DOIUrl":"https://doi.org/10.1155/2023/4469908","url":null,"abstract":"Neutrosophic logic is frequently applied to the engineering technology, scientific administration, and financial matters, among other fields. In addition, neutrosophic linear systems can be used to illustrate various practical problems. Due to the complexity of neutrosophic operators, however, solving linear neutrosophic systems is challenging. This work proposes a new straightforward method for solving the neutrosophic system of linear equations based on the neutrosophic structured element (NSE). Here unknown and right-hand side vectors are considered as triangular neutrosophic numbers. Based on the NSE, analytical expressions of the solution to this equation and its degrees are also provided. Finally, several examples of the methodology are provided.","PeriodicalId":49111,"journal":{"name":"Advances in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45000973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sharmin Akter, M. D. Hossain, M. F. Uddin, M. Hafez
This article deals with the basic features of collisional radial displacements in a prestressed thin elastic tube filled having inviscid fluid with the presence of nonlocal operator. By implementing the extended Poincare–Lighthill–Kuo method and a variational approach, the new two-sided beta time fractional Korteweg-de-Vries (BTF-KdV) equations are derived based on the concept of beta fractional derivative (BFD). Additionally, the BTF-KdV equations are suggested to observe the effect of related parameters on the local and nonlocal coherent head-on collision phenomena for the considered system. It is observed that the proposed equations along with their new solutions not only applicable with the presence of locality but also nonlocality to study the resonance wave phenomena in fluid-filled elastic tube. The outcomes reveal that the BFD and other physical parameters related to tube and fluid have a significant impact on the propagation of pressure wave structures.
{"title":"Collisional Solitons Described by Two-Sided Beta Time Fractional Korteweg-de Vries Equations in Fluid-Filled Elastic Tubes","authors":"Sharmin Akter, M. D. Hossain, M. F. Uddin, M. Hafez","doi":"10.1155/2023/9594339","DOIUrl":"https://doi.org/10.1155/2023/9594339","url":null,"abstract":"This article deals with the basic features of collisional radial displacements in a prestressed thin elastic tube filled having inviscid fluid with the presence of nonlocal operator. By implementing the extended Poincare–Lighthill–Kuo method and a variational approach, the new two-sided beta time fractional Korteweg-de-Vries (BTF-KdV) equations are derived based on the concept of beta fractional derivative (BFD). Additionally, the BTF-KdV equations are suggested to observe the effect of related parameters on the local and nonlocal coherent head-on collision phenomena for the considered system. It is observed that the proposed equations along with their new solutions not only applicable with the presence of locality but also nonlocality to study the resonance wave phenomena in fluid-filled elastic tube. The outcomes reveal that the BFD and other physical parameters related to tube and fluid have a significant impact on the propagation of pressure wave structures.","PeriodicalId":49111,"journal":{"name":"Advances in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43155613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study constant mean curvature hypersurfaces constructed over the fiber of warped products . In this setting, assuming that the sign of the angle function does not changed along the hypersurfaces, we infer the uniqueness of such hypersurfaces by applying a parabolicity criterion. As an application, we get some Bernstein type theorems.
我们研究了在弯曲产物I ×的纤维mn上构造的常平均曲率超曲面f M n。在这种情况下，假设角度函数的符号沿着超曲面不改变，我们通过应用抛物线性准则来推断这种超曲面的唯一性。作为应用，我们得到了一些Bernstein型定理。
{"title":"On Bernstein’s Problem of Complete Parabolic Hypersurfaces in Warped Products","authors":"Ning Zhang, Zhangsheng Zhu","doi":"10.1155/2023/6390463","DOIUrl":"https://doi.org/10.1155/2023/6390463","url":null,"abstract":"<jats:p>We study constant mean curvature hypersurfaces constructed over the fiber <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <msup>\u0000 <mrow>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msup>\u0000 </math>\u0000 </jats:inline-formula> of warped products <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\u0000 <mi>I</mi>\u0000 <msub>\u0000 <mrow>\u0000 <mo>×</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </msub>\u0000 <msup>\u0000 <mrow>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msup>\u0000 </math>\u0000 </jats:inline-formula>. In this setting, assuming that the sign of the angle function does not changed along the hypersurfaces, we infer the uniqueness of such hypersurfaces by applying a parabolicity criterion. As an application, we get some Bernstein type theorems.</jats:p>","PeriodicalId":49111,"journal":{"name":"Advances in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48439660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Mohammadi, A. R. Vahidi, T. Damercheli, S. Khezerloo
In the present work, a new approximated method for solving the nonlinear Duffing-Van der Pol (D-VdP) oscillator equation is suggested. The approximate solution of this equation is introduced with two separate techniques. First, we convert nonlinear D-VdP equation to a nonlinear Volterra integral equation of the second kind (VIESK) using integration, and then, we approximate it with the hybrid Legendre polynomials and block-pulse function (HLBPFs). The next technique is to convert this equation into a system of ordinary differential equation of the first order (SODE) and solve it according to the proposed approximate method. The main goal of the presented technique is to transform these problems into a nonlinear system of algebraic equations using the operational matrix obtained from the integration, which can be solved by a proper numerical method; thus, the solution procedures are either reduced or simplified accordingly. The benefit of the hybrid functions is that they can be adjusted for different values of and , in addition to being capable of yield greater correct numerical answers than the piecewise constant orthogonal function, for the results of integral equations. Resolved governance equation using the Runge-Kutta fourth order algorithm with the stepping time 0.01 s via numerical solution. The approximate results obtained from the proposed method show that this method is effective. The evaluation has been proven that the proposed technique is in good agreement with the numerical results of other methods.
本文提出了求解非线性Duffing-Van der Pol (D-VdP)振子方程的一种新的近似方法。用两种不同的方法给出了该方程的近似解。首先利用积分法将非线性D-VdP方程转化为第二类非线性Volterra积分方程(VIESK)，然后利用混合Legendre多项式和块脉冲函数(HLBPFs)对其进行近似。下一项技术是将该方程转化为一阶常微分方程(SODE)，并根据所提出的近似方法求解。该方法的主要目的是利用由积分得到的运算矩阵将这些问题转化为非线性代数方程组，并用适当的数值方法求解;因此，求解过程相应地减少或简化。混合函数的好处是，它们可以根据n和m的不同值进行调整，此外，对于积分方程的结果，除了能够产生比分段常数正交函数更正确的数值答案外，还能得到更正确的数值答案。采用步进时间为0.01 s的龙格-库塔四阶算法，通过数值求解求解治理方程。近似结果表明，该方法是有效的。计算结果表明，该方法与其他方法的数值计算结果吻合较好。
{"title":"Numerical Solutions of Duffing Van der Pol Equations on the Basis of Hybrid Functions","authors":"M. Mohammadi, A. R. Vahidi, T. Damercheli, S. Khezerloo","doi":"10.1155/2023/4144552","DOIUrl":"https://doi.org/10.1155/2023/4144552","url":null,"abstract":"In the present work, a new approximated method for solving the nonlinear Duffing-Van der Pol (D-VdP) oscillator equation is suggested. The approximate solution of this equation is introduced with two separate techniques. First, we convert nonlinear D-VdP equation to a nonlinear Volterra integral equation of the second kind (VIESK) using integration, and then, we approximate it with the hybrid Legendre polynomials and block-pulse function (HLBPFs). The next technique is to convert this equation into a system of ordinary differential equation of the first order (SODE) and solve it according to the proposed approximate method. The main goal of the presented technique is to transform these problems into a nonlinear system of algebraic equations using the operational matrix obtained from the integration, which can be solved by a proper numerical method; thus, the solution procedures are either reduced or simplified accordingly. The benefit of the hybrid functions is that they can be adjusted for different values of <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\"> <mi>n</mi> </math> and <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\"> <mi>m</mi> </math> , in addition to being capable of yield greater correct numerical answers than the piecewise constant orthogonal function, for the results of integral equations. Resolved governance equation using the Runge-Kutta fourth order algorithm with the stepping time 0.01 s via numerical solution. The approximate results obtained from the proposed method show that this method is effective. The evaluation has been proven that the proposed technique is in good agreement with the numerical results of other methods.","PeriodicalId":49111,"journal":{"name":"Advances in Mathematical Physics","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135558585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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