This paper is dedicated to implementing and presenting numerical algorithms for solving some linear and nonlinear even-order two-point boundary value problems.For this purpose,we establish new explicit formulas for th...This paper is dedicated to implementing and presenting numerical algorithms for solving some linear and nonlinear even-order two-point boundary value problems.For this purpose,we establish new explicit formulas for the high-order derivatives of certain two classes of Jacobi polynomials in terms of their corresponding Jacobi polynomials.These two classes generalize the two celebrated non-symmetric classes of polynomials,namely,Chebyshev polynomials of third-and fourth-kinds.The idea of the derivation of such formulas is essentially based on making use of the power series representations and inversion formulas of these classes of polynomials.The derived formulas serve in converting the even-order linear differential equations with their boundary conditions into linear systems that can be efficiently solved.Furthermore,and based on the first-order derivatives formula of certain Jacobi polynomials,the operational matrix of derivatives is extracted and employed to present another algorithm to treat both linear and nonlinear two-point boundary value problems based on the application of the collocation method.Convergence analysis of the proposed expansions is investigated.Some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.展开更多
Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find on approximation to f(t) by the use of the dassical Jacobi polynomials. The main contribution of our work is the development of a...Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find on approximation to f(t) by the use of the dassical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series ex-pansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.展开更多
We construct photon-subtracted(-added)thermo vacuum state by normalizing them.As their application we derive some new generating function formulas of Jacobi polynomials,which may be applied to study other problems in ...We construct photon-subtracted(-added)thermo vacuum state by normalizing them.As their application we derive some new generating function formulas of Jacobi polynomials,which may be applied to study other problems in quantum mechanics.This will also stimulate the research of mathematical physics in the future.展开更多
Tw o variable Jacobi polynomials,as a two-dimensional basis,are applied to solve a class of temporal fractional partial differential equations.The fractional derivative operators are in the Caputo sense.The operationa...Tw o variable Jacobi polynomials,as a two-dimensional basis,are applied to solve a class of temporal fractional partial differential equations.The fractional derivative operators are in the Caputo sense.The operational matrices of the integration of integer and fractional orders are presented.Using these matrices together with the Tau Jacobi method converts the main problem into the corresponding system of algebraic equations.An error bound is obtained in a two-dimensional Jacobi-weighted Sobolev space.Finally,the efficiency of the proposed method is demonstrated by implementing the algorithm to several illustrative examples.Results will be compared witli those obtained from some existing methods.展开更多
We use the constituent quark model to extract polarized parton distributions and finally polarized nucleon structure function.Due to limited experimental data which do not cover whole （x,Q 2 ） plane and to increase ...We use the constituent quark model to extract polarized parton distributions and finally polarized nucleon structure function.Due to limited experimental data which do not cover whole （x,Q 2 ） plane and to increase the reliability of the fitting,we employ the Jacobi orthogonal polynomials expansion.It will be possible to extract the polarized structure functions for Helium,using the convolution of the nucleon polarized structure functions with the light cone moment distribution.The results are in good agreement with available experimental data and some theoretical models.展开更多
Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting proper...Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.展开更多
Many fields require the zeros of orthogonal polynomials. In this paper, the middle variable was improved to give a new asymptotic approximation, with error bounds, for the Jacobi polynomials P (α,β) n(cosθ) (...Many fields require the zeros of orthogonal polynomials. In this paper, the middle variable was improved to give a new asymptotic approximation, with error bounds, for the Jacobi polynomials P (α,β) n(cosθ) (0≤θ≤π/2,α,β>-1), as n→+∞. An accurate approximation with error bounds is also constructed for the zero θ n,s of P (α,β) n(cosθ)(α≥0,β>-1).展开更多
The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polyn...The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of operator disentanglement some operator identities are derived in an appendix. They allow calculating convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions.展开更多
We consider the problem of estimating the derivative of a function f from its noisy version fδby using the derivatives of the partial sums of Fourier-Legendre series of f. Instead of the observation Lspace, we perfor...We consider the problem of estimating the derivative of a function f from its noisy version fδby using the derivatives of the partial sums of Fourier-Legendre series of f. Instead of the observation Lspace, we perform the reconstruction of the derivative in a weighted Lspace. This takes full advantage of the properties of Legendre polynomials and results in a slight improvement on the convergence order. Finally, we provide several numerical examples to demonstrate the efficiency of the proposed method.展开更多
In this article, we develop a fully Discrete Galerkin（DG） method for solving ini- tial value fractional integro-differential equations（FIDEs）. We consider Generalized Jacobi polynomials（CJPs） with indexes corres...In this article, we develop a fully Discrete Galerkin（DG） method for solving ini- tial value fractional integro-differential equations（FIDEs）. We consider Generalized Jacobi polynomials（CJPs） with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution. Numerical results are presented to demonstrate the effectiveness of the proposed method.展开更多
The solutions of the Laplace equation in n-dimensional space are studied. The angular eigenfunctions have the form of associated Jacob/polynomials. The radial solution of the Helmholtz equation is derived.
For the formal presentation about the definite problems of ultra-hyperbolic equations, the famous Asgeirsson mean value theorem has answered that Cauchy problems are ill-posed to ultra-hyperbolic partial differential ...For the formal presentation about the definite problems of ultra-hyperbolic equations, the famous Asgeirsson mean value theorem has answered that Cauchy problems are ill-posed to ultra-hyperbolic partial differential equations of the second-order. So it is important to develop Asgeirsson mean value theorem. The mean value of solution for the higher order equation hay been discussed primarily and has no exact result at present. The mean value theorem for the higher order equation can be deduced and satisfied generalized biaxial symmetry potential equation by using the result of Asgeirsson mean value theorem and the properties of derivation and integration. Moreover, the mean value formula can be obtained by using the regular solutions of potential equation and the special properties of Jacobi polynomials. Its converse theorem is also proved. The obtained results make it possible to discuss on continuation of the solutions and well posed problem.展开更多
In this paper, we have considered the generalized bi-axially symmetric SchrSdinger equation ■2φ/■x2 ＋ ■2φ/■y2＋2v/x ■φ/■x＋2μ/y ■φ/■y＋{K2-V（r）}φ=0 ,where μ, v ≥ 0, and rV（r） is an entire function...In this paper, we have considered the generalized bi-axially symmetric SchrSdinger equation ■2φ/■x2 ＋ ■2φ/■y2＋2v/x ■φ/■x＋2μ/y ■φ/■y＋{K2-V（r）}φ=0 ,where μ, v ≥ 0, and rV（r） is an entire function of r=＋（s2＋y2）1/2 corresponding to a scattering potential V（r）. Growth paramet ers of entire function solutions in terms of their expansion coefficients, which are analogous to the formulas for order and type occurring in classical function theory, have been obtained. Our results are applicable for the scattering of particles in quantum mechanics.展开更多
The paper deals with growth estimates and approximation（not necessarily entire） of solutions of certain elliptic partial differential equations. These solutions are called generalized bi-axially symmetric potentials...The paper deals with growth estimates and approximation（not necessarily entire） of solutions of certain elliptic partial differential equations. These solutions are called generalized bi-axially symmetric potentials（GBASP＇s）. To obtain more refined measure of growth, we have defined q-proximate order and obtained the characterization of generalized q-type and generalized lower q-type with respect to q-proximate order of a GBASP in terms of approximation errors and ratio of these errors in sup norm.展开更多
This paper proposes and applies a method to sort two-dimensional control points of triangular Bezier surfaces in a row vector. Using the property of bivariate Jacobi basis functions, it further presents two algorithms...This paper proposes and applies a method to sort two-dimensional control points of triangular Bezier surfaces in a row vector. Using the property of bivariate Jacobi basis functions, it further presents two algorithms for multi-degree reduction of triangular Bezier surfaces with constraints, providing explicit degree-reduced surfaces. The first algorithm can obtain the explicit representation of the optimal degree-reduced surfaces and the approximating error in both boundary curve constraints and corner constraints. But it has to solve the inversion of a matrix whose degree is related with the original surface. The second algorithm entails no matrix inversion to bring about computational instability, gives stable degree-reduced surfaces quickly, and presents the error bound. In the end, the paper proves the efficiency of the two algorithms through examples and error analysis.展开更多
For the first time, we derive the photon number cumulant for two-mode squeezed state and show that its cumulant expansion leads to normalization of two-mode photon subtracted-squeezed states and photon added- squeezed...For the first time, we derive the photon number cumulant for two-mode squeezed state and show that its cumulant expansion leads to normalization of two-mode photon subtracted-squeezed states and photon added- squeezed states. We show that the normalization is related to Jacobi polynomial, so the cumulant expansion in turn represents the new generating function of Jacobi polynomial.展开更多
In this paper,we study the Jacobi frame approximation with equispaced samples and derive an error estimation.We observe numerically that the approximation accuracy gradually decreases as the extended domain parameter ...In this paper,we study the Jacobi frame approximation with equispaced samples and derive an error estimation.We observe numerically that the approximation accuracy gradually decreases as the extended domain parameter γ increases in the uniform norm,especially for differentiable functions.In addition,we show that when the indexes of Jacobi polynomials α and β are larger(for example max{α,β}>10),it leads to a divergence behavior on the frame approximation error decay.展开更多
A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a map...A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron, and can be taken as an extension of the 2-D Steiner domain. The polynomials can be viewed as Jacobi polynomials on such a domain. Three-term relations are derived explicitly. The number of the individual terms, involved in the recurrences relations, are shown to be independent on the total degree of the polynomials. The numbers now are determined to be five and seven, with respect to two conjugate variables z, $ \bar z $ and a real variable r, respectively. Three examples are discussed in details, which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds, and Legendre polynomials.展开更多
In this research work,we study the Human Immunodeficiency Virus(HIV)infection on helper T cells governed by a mathematical model consisting of a system of three first-order nonlinear differential equations.The objecti...In this research work,we study the Human Immunodeficiency Virus(HIV)infection on helper T cells governed by a mathematical model consisting of a system of three first-order nonlinear differential equations.The objective of the analysis is to present an approximate mathematical solution to the model that gives the count of the numbers of uninfected and infected helper T cells and the number of free virus particles present at a given instant of time.The system of nonlinear ODEs is converted into a system of nonlinear algebraic equations using spectral collocation method with three different basis functions such as Chebyshev,Legendre and Jacobi polynomials.Some factors such as the production of helper T cells and infection of these cells play a vital role in infected and uninfected cell counts.Detailed error analysis is done to compare our results with the existing methods.It is shown that the spectral collocation method is a very reliable,efficient and robust method of solution compared to many other solution procedures available in the literature.All these results are presented in the form of tables and figures.展开更多
Necessary and sufficient conditions for the regularity and q-regularity of (0,1,…,m-2,m)interpolation on the zeros of (1-x2)P_(n-2)￣(α,β)(x) (α,β>-1) in a manageable form are established,where P_(n-2)￣(α,β...Necessary and sufficient conditions for the regularity and q-regularity of (0,1,…,m-2,m)interpolation on the zeros of (1-x2)P_(n-2)￣(α,β)(x) (α,β>-1) in a manageable form are established,where P_(n-2)￣(α,β)(x) stands for the (n-2)th Jacobi polynomial. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, we show that under a mild assumption if the problem of (0,1,…,m-2,m) interpolation has an infinity of solutions then the general form of the solutions is fo(x)+Cf(x) with an arbitrary constant C.展开更多
文摘This paper is dedicated to implementing and presenting numerical algorithms for solving some linear and nonlinear even-order two-point boundary value problems.For this purpose,we establish new explicit formulas for the high-order derivatives of certain two classes of Jacobi polynomials in terms of their corresponding Jacobi polynomials.These two classes generalize the two celebrated non-symmetric classes of polynomials,namely,Chebyshev polynomials of third-and fourth-kinds.The idea of the derivation of such formulas is essentially based on making use of the power series representations and inversion formulas of these classes of polynomials.The derived formulas serve in converting the even-order linear differential equations with their boundary conditions into linear systems that can be efficiently solved.Furthermore,and based on the first-order derivatives formula of certain Jacobi polynomials,the operational matrix of derivatives is extracted and employed to present another algorithm to treat both linear and nonlinear two-point boundary value problems based on the application of the collocation method.Convergence analysis of the proposed expansions is investigated.Some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.
文摘Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find on approximation to f(t) by the use of the dassical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series ex-pansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.
基金Doctoral Scientific Research Foundation of Chaohu College(No.KYQD-201407)
文摘We construct photon-subtracted(-added)thermo vacuum state by normalizing them.As their application we derive some new generating function formulas of Jacobi polynomials,which may be applied to study other problems in quantum mechanics.This will also stimulate the research of mathematical physics in the future.
基金the Iran National Science Foundation:INFS under Grant No.95009788 and is also under supplementary support of The University of Guilan,Iran.
文摘Tw o variable Jacobi polynomials,as a two-dimensional basis,are applied to solve a class of temporal fractional partial differential equations.The fractional derivative operators are in the Caputo sense.The operational matrices of the integration of integer and fractional orders are presented.Using these matrices together with the Tau Jacobi method converts the main problem into the corresponding system of algebraic equations.An error bound is obtained in a two-dimensional Jacobi-weighted Sobolev space.Finally,the efficiency of the proposed method is demonstrated by implementing the algorithm to several illustrative examples.Results will be compared witli those obtained from some existing methods.
文摘We use the constituent quark model to extract polarized parton distributions and finally polarized nucleon structure function.Due to limited experimental data which do not cover whole （x,Q 2 ） plane and to increase the reliability of the fitting,we employ the Jacobi orthogonal polynomials expansion.It will be possible to extract the polarized structure functions for Helium,using the convolution of the nucleon polarized structure functions with the light cone moment distribution.The results are in good agreement with available experimental data and some theoretical models.
文摘Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.
基金Supported by the Natural Science Foundation of Beijing
文摘Many fields require the zeros of orthogonal polynomials. In this paper, the middle variable was improved to give a new asymptotic approximation, with error bounds, for the Jacobi polynomials P (α,β) n(cosθ) (0≤θ≤π/2,α,β>-1), as n→+∞. An accurate approximation with error bounds is also constructed for the zero θ n,s of P (α,β) n(cosθ)(α≥0,β>-1).
文摘The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of operator disentanglement some operator identities are derived in an appendix. They allow calculating convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions.
基金Supported by the National Nature Science Foundation of China(Grant Nos.1130105211301045+4 种基金114010771127106011290143)the Fundamental Research Funds for the Central Universities(Grant No.DUT15RC(3)058)the Fundamental Research of Civil Aircraft(Grant No.MJ-F-2012-04)
文摘We consider the problem of estimating the derivative of a function f from its noisy version fδby using the derivatives of the partial sums of Fourier-Legendre series of f. Instead of the observation Lspace, we perform the reconstruction of the derivative in a weighted Lspace. This takes full advantage of the properties of Legendre polynomials and results in a slight improvement on the convergence order. Finally, we provide several numerical examples to demonstrate the efficiency of the proposed method.
文摘In this article, we develop a fully Discrete Galerkin（DG） method for solving ini- tial value fractional integro-differential equations（FIDEs）. We consider Generalized Jacobi polynomials（CJPs） with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution. Numerical results are presented to demonstrate the effectiveness of the proposed method.
基金Supported by the Nationa1 Natural Science Foundation of China under Grant No.10874018"the Fundamental Research Funds for the Central Universities"
文摘The solutions of the Laplace equation in n-dimensional space are studied. The angular eigenfunctions have the form of associated Jacob/polynomials. The radial solution of the Helmholtz equation is derived.
文摘For the formal presentation about the definite problems of ultra-hyperbolic equations, the famous Asgeirsson mean value theorem has answered that Cauchy problems are ill-posed to ultra-hyperbolic partial differential equations of the second-order. So it is important to develop Asgeirsson mean value theorem. The mean value of solution for the higher order equation hay been discussed primarily and has no exact result at present. The mean value theorem for the higher order equation can be deduced and satisfied generalized biaxial symmetry potential equation by using the result of Asgeirsson mean value theorem and the properties of derivation and integration. Moreover, the mean value formula can be obtained by using the regular solutions of potential equation and the special properties of Jacobi polynomials. Its converse theorem is also proved. The obtained results make it possible to discuss on continuation of the solutions and well posed problem.
文摘In this paper, we have considered the generalized bi-axially symmetric SchrSdinger equation ■2φ/■x2 ＋ ■2φ/■y2＋2v/x ■φ/■x＋2μ/y ■φ/■y＋{K2-V（r）}φ=0 ,where μ, v ≥ 0, and rV（r） is an entire function of r=＋（s2＋y2）1/2 corresponding to a scattering potential V（r）. Growth paramet ers of entire function solutions in terms of their expansion coefficients, which are analogous to the formulas for order and type occurring in classical function theory, have been obtained. Our results are applicable for the scattering of particles in quantum mechanics.
文摘The paper deals with growth estimates and approximation（not necessarily entire） of solutions of certain elliptic partial differential equations. These solutions are called generalized bi-axially symmetric potentials（GBASP＇s）. To obtain more refined measure of growth, we have defined q-proximate order and obtained the characterization of generalized q-type and generalized lower q-type with respect to q-proximate order of a GBASP in terms of approximation errors and ratio of these errors in sup norm.
基金Supported by the National Natural Science Foundation of China (6087311160933007)
文摘This paper proposes and applies a method to sort two-dimensional control points of triangular Bezier surfaces in a row vector. Using the property of bivariate Jacobi basis functions, it further presents two algorithms for multi-degree reduction of triangular Bezier surfaces with constraints, providing explicit degree-reduced surfaces. The first algorithm can obtain the explicit representation of the optimal degree-reduced surfaces and the approximating error in both boundary curve constraints and corner constraints. But it has to solve the inversion of a matrix whose degree is related with the original surface. The second algorithm entails no matrix inversion to bring about computational instability, gives stable degree-reduced surfaces quickly, and presents the error bound. In the end, the paper proves the efficiency of the two algorithms through examples and error analysis.
基金Project supported by the Natural Science Foundation of Fujian Province,China (Grant No.2011J01018)the National Natural Science Foundation of China (Grant No.11175113)
文摘For the first time, we derive the photon number cumulant for two-mode squeezed state and show that its cumulant expansion leads to normalization of two-mode photon subtracted-squeezed states and photon added- squeezed states. We show that the normalization is related to Jacobi polynomial, so the cumulant expansion in turn represents the new generating function of Jacobi polynomial.
文摘In this paper,we study the Jacobi frame approximation with equispaced samples and derive an error estimation.We observe numerically that the approximation accuracy gradually decreases as the extended domain parameter γ increases in the uniform norm,especially for differentiable functions.In addition,we show that when the indexes of Jacobi polynomials α and β are larger(for example max{α,β}>10),it leads to a divergence behavior on the frame approximation error decay.
基金the Major Basic Project of China(Grant No.2005CB321702)the National Natural Science Foundation of China(Grant Nos.10431050,60573023)
文摘A new class of three-variable orthogonal polynomials, defined as eigenfunctions of a second order PDE operator, is studied. These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron, and can be taken as an extension of the 2-D Steiner domain. The polynomials can be viewed as Jacobi polynomials on such a domain. Three-term relations are derived explicitly. The number of the individual terms, involved in the recurrences relations, are shown to be independent on the total degree of the polynomials. The numbers now are determined to be five and seven, with respect to two conjugate variables z, $ \bar z $ and a real variable r, respectively. Three examples are discussed in details, which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds, and Legendre polynomials.
文摘In this research work,we study the Human Immunodeficiency Virus(HIV)infection on helper T cells governed by a mathematical model consisting of a system of three first-order nonlinear differential equations.The objective of the analysis is to present an approximate mathematical solution to the model that gives the count of the numbers of uninfected and infected helper T cells and the number of free virus particles present at a given instant of time.The system of nonlinear ODEs is converted into a system of nonlinear algebraic equations using spectral collocation method with three different basis functions such as Chebyshev,Legendre and Jacobi polynomials.Some factors such as the production of helper T cells and infection of these cells play a vital role in infected and uninfected cell counts.Detailed error analysis is done to compare our results with the existing methods.It is shown that the spectral collocation method is a very reliable,efficient and robust method of solution compared to many other solution procedures available in the literature.All these results are presented in the form of tables and figures.
文摘Necessary and sufficient conditions for the regularity and q-regularity of (0,1,…,m-2,m)interpolation on the zeros of (1-x2)P_(n-2)￣(α,β)(x) (α,β>-1) in a manageable form are established,where P_(n-2)￣(α,β)(x) stands for the (n-2)th Jacobi polynomial. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, we show that under a mild assumption if the problem of (0,1,…,m-2,m) interpolation has an infinity of solutions then the general form of the solutions is fo(x)+Cf(x) with an arbitrary constant C.
