A novel method for the determination of vitamin C（Vc） is proposed in this article. After the reaction with Folin-Ciocalteau reagent at ambient temperature, Vc solution was scanned at 750--1100 nm, and its first-orde...A novel method for the determination of vitamin C（Vc） is proposed in this article. After the reaction with Folin-Ciocalteau reagent at ambient temperature, Vc solution was scanned at 750--1100 nm, and its first-order derivative spectrum were obtained from the original spectrum. The values of derivative selected at 995 nm were used for determination. It was proved that Vc could quickly react with Folin-Ciocalteau reagent within 5 min and the product was quite stable for a long time. The conditions required for this method is not very complicated, its precision and accuracy are similar to those of the iodometric titration described in Chinese Pharmacopoeia, and the limit of detection is 0.312 μg/mL. The determination of the results of vitamin C tablet, pill, and injection demonstrates that this method has wide pharmaceutical applications.展开更多
Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different application...Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the(n-1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached.展开更多
Importance analysis quantifies the critical degree of individual component. Compared with the traditional binary state system,importance analysis of the multi-state system is more aligned with the practice. Because th...Importance analysis quantifies the critical degree of individual component. Compared with the traditional binary state system,importance analysis of the multi-state system is more aligned with the practice. Because the multi-valued decision diagram( MDD) can reflect the relationship between the components and the system state bilaterally, it was introduced into the reliability calculation of the multi-state system( MSS). The building method,simplified criteria,and path search and probability algorithm of MSS structure function MDD were given,and the reliability of the system was calculated. The computing methods of importance based on MDD and direct partial logic derivatives( DPLD) were presented. The diesel engine fuel supply system was taken as an example to illustrate the proposed method. The results show that not only the probability of the system in each state can be easily obtained,but also the influence degree of each component and its state on the system reliability can be obtained,which is conducive to the condition monitoring and structure optimization of the system.展开更多
It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment...It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular. The method consists to solve the integral equation numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.展开更多
An L-stable block method based on hybrid second derivative algorithm (BHSDA) is provided by a continuous second derivative method that is defined for all values of the independent variable and applied to parabolic par...An L-stable block method based on hybrid second derivative algorithm (BHSDA) is provided by a continuous second derivative method that is defined for all values of the independent variable and applied to parabolic partial differential equations (PDEs). The use of the BHSDA to solve PDEs is facilitated by the method of lines which involves making an approximation to the space derivatives, and hence reducing the problem to that of solving a time-dependent system of first order initial value ordinary differential equations. The stability properties of the method is examined and some numerical results presented.展开更多
The classical heat conduction equation is derived from the assumption that the temperature increases immediately after heat transfer, but the increase of temperature is a slow process, so the memory-dependent heat con...The classical heat conduction equation is derived from the assumption that the temperature increases immediately after heat transfer, but the increase of temperature is a slow process, so the memory-dependent heat conduction model has been reconstructed. Numerical results show that the solution of the initial boundary value problem of the new model is similar to that of the classical heat conduction equation, but its propagation speed is slower than that of the latter. In addition, the propagation speed of the former is also affected by time delay and kernel function.展开更多
This paper reports the results on the nature of bond-order and net charge distributions predicted by Ab initio Hartree- Fock procedures for 1-amino-2-iminio-, 1-amino-3-iminio- and 1-amino-4-iminiotropylium cations th...This paper reports the results on the nature of bond-order and net charge distributions predicted by Ab initio Hartree- Fock procedures for 1-amino-2-iminio-, 1-amino-3-iminio- and 1-amino-4-iminiotropylium cations that incorporate, in order, the 1,7-, 1,3- and 1,5-diazapentadienium (vinamidinium) elements. There appears to be very little contribution from tropylium-type charge distribution, the positive charges residing largely in the nitrogen atoms. The partial bond fixations and charge distributions show interesting variation in the three isomers. The 1,3-isomer in which the 1,3-diazapentadienium element is preserved in the favoured zigzag conformation appears to be relatively the best stabilized. The six isomeric benzo-fused derivatives arising from the three amino-iminiotropylium cations show similar differences in patterns of behaviour. Interestingly, the isomer in which a zigzag 1,3-diazapentadienium element is conjugated with a styrene moiety receives the deepest stabilization. While showing that the element largely contributes to the relative stabilization among the systems studied, contribution from certain stereochemical destabilizing factors may not be insignificant.展开更多
We present here a high-order numerical formula for approximating the Caputo fractional derivative of order𝛼for 0<α<1.This new formula is on the basis of the third degree Lagrange interpolating polynomia...We present here a high-order numerical formula for approximating the Caputo fractional derivative of order𝛼for 0<α<1.This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial diff erential equations.In comparison with the previous formulae,the main superiority of the new formula is its order of accuracy which is 4−α,while the order of accuracy of the previous ones is less than 3.It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost.The eff ectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples.Moreover,an application of the new formula in solving some fractional partial diff erential equations is presented by constructing a fi nite diff erence scheme.A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.展开更多
Near-infrared (NIR) spectroscopy combined with chemometrics methods was applied to the rapid and reagent-free analysis of serum urea nitrogen (SUN). The mul-partitions modeling was performed to achieve parameter stabi...Near-infrared (NIR) spectroscopy combined with chemometrics methods was applied to the rapid and reagent-free analysis of serum urea nitrogen (SUN). The mul-partitions modeling was performed to achieve parameter stability. A large-scale parameter cyclic and global optimization platform for Norris derivative filter (NDF) of three parameters (the derivative order: d, the number of smoothing points: s and the number of differential gaps: g) was developed with PLS regression. Meantime, the parameters’ adaptive analysis of NDF algorithm was also given, and achieved a significantly better modeling effect than one without spectral pre-processing. After eliminating the interference wavebands of saturated absorption, the modeling performance was further improved. In validation, the root mean square error (SEP), correlation coefficient (RP) for prediction and the ratio of performance to deviation (RPD) were 1.66 mmol?L-1, 0.966 and 4.7, respectively. The results showed that the high-precision analysis of SUN was feasibility based on NIR spectroscopy and Norris-PLS. The global optimization method of NDF is also expected to be applied to other analysis objects.展开更多
Motivated by the widely used ans¨atz method and starting from the modified Riemann–Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional pa...Motivated by the widely used ans¨atz method and starting from the modified Riemann–Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.展开更多
We introduce a novel numerical method for solving two-sided space fractional partial differential equations in two-dimensional case.The approximation of the space fractional Riemann-Liouville derivative is based on th...We introduce a novel numerical method for solving two-sided space fractional partial differential equations in two-dimensional case.The approximation of the space fractional Riemann-Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order O(h^3-a),where h is the space step size and α∈(1,2)is the order of Riemann-Liouville fractional derivative.Based on this scheme,we introduce a shifted finite difference method for solving space fractional partial differential equations.We obtained the error estimates with the convergence orders O(τ+h^3-a+h^β),where τ is the time step size and β>0 is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation.Unlike the numerical methods for solving space fractional partial differential equations constructed using the standard shifted Griinwald-Letnikov formula or higher order Lubich's methods which require the solution of the equation to satisfy the homogeneous Dirichlet boundary condition to get the firstorder convergence,the numerical method for solving the space fractional partial differential equation constructed using the Hadamard finite-part integral approach does not require the solution of the equation to satisfy the Dirichlet homogeneous boundary condition.Numerical results show that the experimentally determined convergence order obtained using the Hadamard finite-part integral approach for solving the space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained using the numerical methods constructed with the standard shifted Griinwald-Letnikov formula or Lubich's higher order approximation schemes.展开更多
Orthotopic liver transplantation(OLT)is the only proven effective treatment for both end-stage and metabolic liver diseases.Hepatocyte transplantation is a promising alternative for OLT,but the lack of available donor...Orthotopic liver transplantation(OLT)is the only proven effective treatment for both end-stage and metabolic liver diseases.Hepatocyte transplantation is a promising alternative for OLT,but the lack of available donor livers has hampered its clinical application.Hepatocyte-like cells(HLCs)differentiated from many multi-potential stem cells can help repair damaged liver tissue.Yet almost suitable cells currently identified for human use are difficult to harvest and involve invasive procedures.Recently,a novel mesenchymal stem cell derived from human menstrual blood(MenSC)has been discovered and obtained easily and repeatedly.In this study,we examined whether the MenSCs are able to differentiate into functional HLCs in vitro.After three weeks of incubation in hepatogenic differentiation medium containing hepatocyte growth factor(HGF),fibroblast growth factor-4(FGF-4),and oncostain M(OSM),cuboidal HLCs were observed,and cells also expressed hepatocyte-specific marker genes including albumin(ALB),α-fetoprotein(AFP),cytokeratin 18/19(CK18/19),and cytochrome P450 1A1/3A4(CYP1A1/3A4).Differentiated cells further demonstrated in vitro mature hepatocyte functions such as urea synthesis,glycogen storage,and indocyanine green(ICG)uptake.After intrasplenic transplantation into mice with 2/3 partial hepatectomy,the MenSC-derived HLCs were detected in recipient livers and expressed human ALB protein.We also showed that MenSC-derived HLC transplantation could restore the serum ALB level and significantly suppressed transaminase activity of liver injury animals.In conclusion,MenSCs may serve as an ideal,easily accessible source of material for tissue engineering and cell therapy of liver tissues.展开更多
In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptot...In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptotic uniform stability and Mittag Leffler stability.The approach presented is based on the specially introduced piecewise continuous Lyapunov functions.Furthermore,some numerical examples are given to show the effectiveness of our obtained theoretical results.展开更多
In this paper,we mainly investigate the value distribution of meromorphic functions in Cmwith its partial differential and uniqueness problem on meromorphic functions in Cmand with its k-th total derivative sharing sm...In this paper,we mainly investigate the value distribution of meromorphic functions in Cmwith its partial differential and uniqueness problem on meromorphic functions in Cmand with its k-th total derivative sharing small functions.As an application of the value distribution result,we study the defect relation of a nonconstant solution to the partial differential equation.In particular,we give a connection between the Picard type theorem of Milliox-Hayman and the characterization of entire solutions of a partial differential equation.展开更多
In this paper, we shall be concerned with the numerical solution of parabolic equations in one space variable and the time variable t. We expand Taylor series to derive a higher-order approximation for U<sub>t&l...In this paper, we shall be concerned with the numerical solution of parabolic equations in one space variable and the time variable t. We expand Taylor series to derive a higher-order approximation for U<sub>t</sub>. We begin with the simplest model problem, for heat conduction in a uniform medium. For this model problem, an explicit difference method is very straightforward in use, and the analysis of its error is easily accomplished by the use of a maximum principle. As we shall show, however, the numerical solution becomes unstable unless the time step is severely restricted, so we shall go on to consider other, more elaborate, numerical methods which can avoid such a restriction. The additional complication in the numerical calculation is more than offset by the smaller number of time steps needed. We then extend the methods to problems with more general boundary conditions, then to more general linear parabolic equations. Finally, we shall discuss the more difficult problem of the solution of nonlinear equations.展开更多
基金Natural Science Foundation of Jilin Province, China(No.200305502)
文摘A novel method for the determination of vitamin C（Vc） is proposed in this article. After the reaction with Folin-Ciocalteau reagent at ambient temperature, Vc solution was scanned at 750--1100 nm, and its first-order derivative spectrum were obtained from the original spectrum. The values of derivative selected at 995 nm were used for determination. It was proved that Vc could quickly react with Folin-Ciocalteau reagent within 5 min and the product was quite stable for a long time. The conditions required for this method is not very complicated, its precision and accuracy are similar to those of the iodometric titration described in Chinese Pharmacopoeia, and the limit of detection is 0.312 μg/mL. The determination of the results of vitamin C tablet, pill, and injection demonstrates that this method has wide pharmaceutical applications.
基金Authors gratefully acknowledge Ajman University for providing facilities for our research under Grant Ref.No.2019-IRG-HBS-11.
文摘Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the(n-1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached.
基金National Natural Science Foundation of China(No.61164009)the Science and Technology Research Project,Department of Education of Jiangxi Province,China(No.GJJ14420)Natural Science Foundation of Jiangxi Province,China(No.20132BAB206026)
文摘Importance analysis quantifies the critical degree of individual component. Compared with the traditional binary state system,importance analysis of the multi-state system is more aligned with the practice. Because the multi-valued decision diagram( MDD) can reflect the relationship between the components and the system state bilaterally, it was introduced into the reliability calculation of the multi-state system( MSS). The building method,simplified criteria,and path search and probability algorithm of MSS structure function MDD were given,and the reliability of the system was calculated. The computing methods of importance based on MDD and direct partial logic derivatives( DPLD) were presented. The diesel engine fuel supply system was taken as an example to illustrate the proposed method. The results show that not only the probability of the system in each state can be easily obtained,but also the influence degree of each component and its state on the system reliability can be obtained,which is conducive to the condition monitoring and structure optimization of the system.
基金Supported in part by the Chinese Outstanding Youth Science Foundation(69925308)supported by Program for Changjiang Scholars and Innovative Research Team in University
文摘It will be shown that finding solutions from the Poisson and Klein-Gordon equations under Neumann conditions are equivalent to solving an integral equation, which can be treated as a generalized two-dimensional moment problem over a domain that is considered rectangular. The method consists to solve the integral equation numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.
文摘An L-stable block method based on hybrid second derivative algorithm (BHSDA) is provided by a continuous second derivative method that is defined for all values of the independent variable and applied to parabolic partial differential equations (PDEs). The use of the BHSDA to solve PDEs is facilitated by the method of lines which involves making an approximation to the space derivatives, and hence reducing the problem to that of solving a time-dependent system of first order initial value ordinary differential equations. The stability properties of the method is examined and some numerical results presented.
文摘The classical heat conduction equation is derived from the assumption that the temperature increases immediately after heat transfer, but the increase of temperature is a slow process, so the memory-dependent heat conduction model has been reconstructed. Numerical results show that the solution of the initial boundary value problem of the new model is similar to that of the classical heat conduction equation, but its propagation speed is slower than that of the latter. In addition, the propagation speed of the former is also affected by time delay and kernel function.
文摘This paper reports the results on the nature of bond-order and net charge distributions predicted by Ab initio Hartree- Fock procedures for 1-amino-2-iminio-, 1-amino-3-iminio- and 1-amino-4-iminiotropylium cations that incorporate, in order, the 1,7-, 1,3- and 1,5-diazapentadienium (vinamidinium) elements. There appears to be very little contribution from tropylium-type charge distribution, the positive charges residing largely in the nitrogen atoms. The partial bond fixations and charge distributions show interesting variation in the three isomers. The 1,3-isomer in which the 1,3-diazapentadienium element is preserved in the favoured zigzag conformation appears to be relatively the best stabilized. The six isomeric benzo-fused derivatives arising from the three amino-iminiotropylium cations show similar differences in patterns of behaviour. Interestingly, the isomer in which a zigzag 1,3-diazapentadienium element is conjugated with a styrene moiety receives the deepest stabilization. While showing that the element largely contributes to the relative stabilization among the systems studied, contribution from certain stereochemical destabilizing factors may not be insignificant.
文摘We present here a high-order numerical formula for approximating the Caputo fractional derivative of order𝛼for 0<α<1.This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial diff erential equations.In comparison with the previous formulae,the main superiority of the new formula is its order of accuracy which is 4−α,while the order of accuracy of the previous ones is less than 3.It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost.The eff ectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples.Moreover,an application of the new formula in solving some fractional partial diff erential equations is presented by constructing a fi nite diff erence scheme.A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.
文摘Near-infrared (NIR) spectroscopy combined with chemometrics methods was applied to the rapid and reagent-free analysis of serum urea nitrogen (SUN). The mul-partitions modeling was performed to achieve parameter stability. A large-scale parameter cyclic and global optimization platform for Norris derivative filter (NDF) of three parameters (the derivative order: d, the number of smoothing points: s and the number of differential gaps: g) was developed with PLS regression. Meantime, the parameters’ adaptive analysis of NDF algorithm was also given, and achieved a significantly better modeling effect than one without spectral pre-processing. After eliminating the interference wavebands of saturated absorption, the modeling performance was further improved. In validation, the root mean square error (SEP), correlation coefficient (RP) for prediction and the ratio of performance to deviation (RPD) were 1.66 mmol?L-1, 0.966 and 4.7, respectively. The results showed that the high-precision analysis of SUN was feasibility based on NIR spectroscopy and Norris-PLS. The global optimization method of NDF is also expected to be applied to other analysis objects.
基金Supported by National Natural Science Foundation of China under Grant Nos.11071278,111471004the Fundamental Research Funds for the Central Universities of GK201302026 and GK201102007
文摘Motivated by the widely used ans¨atz method and starting from the modified Riemann–Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper.
基金Y.Wang's research was supported by the Natural Science Foundation of Luliang University(XN201510).
文摘We introduce a novel numerical method for solving two-sided space fractional partial differential equations in two-dimensional case.The approximation of the space fractional Riemann-Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order O(h^3-a),where h is the space step size and α∈(1,2)is the order of Riemann-Liouville fractional derivative.Based on this scheme,we introduce a shifted finite difference method for solving space fractional partial differential equations.We obtained the error estimates with the convergence orders O(τ+h^3-a+h^β),where τ is the time step size and β>0 is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation.Unlike the numerical methods for solving space fractional partial differential equations constructed using the standard shifted Griinwald-Letnikov formula or higher order Lubich's methods which require the solution of the equation to satisfy the homogeneous Dirichlet boundary condition to get the firstorder convergence,the numerical method for solving the space fractional partial differential equation constructed using the Hadamard finite-part integral approach does not require the solution of the equation to satisfy the Dirichlet homogeneous boundary condition.Numerical results show that the experimentally determined convergence order obtained using the Hadamard finite-part integral approach for solving the space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained using the numerical methods constructed with the standard shifted Griinwald-Letnikov formula or Lubich's higher order approximation schemes.
基金Project supported by the National High-Tech R&D Program(863) of China(Nos.2011AA020102 and 2012AA020905)the Key Technologies R&D Program of Zhejiang Province(Nos.2012C13015-2and 2011C13029-1)+1 种基金the Hangzhou Key Technologies R&D Program(No.20122513A49)the National Natural Science Foundation of China(Nos.81201783 and 81201089)
文摘Orthotopic liver transplantation(OLT)is the only proven effective treatment for both end-stage and metabolic liver diseases.Hepatocyte transplantation is a promising alternative for OLT,but the lack of available donor livers has hampered its clinical application.Hepatocyte-like cells(HLCs)differentiated from many multi-potential stem cells can help repair damaged liver tissue.Yet almost suitable cells currently identified for human use are difficult to harvest and involve invasive procedures.Recently,a novel mesenchymal stem cell derived from human menstrual blood(MenSC)has been discovered and obtained easily and repeatedly.In this study,we examined whether the MenSCs are able to differentiate into functional HLCs in vitro.After three weeks of incubation in hepatogenic differentiation medium containing hepatocyte growth factor(HGF),fibroblast growth factor-4(FGF-4),and oncostain M(OSM),cuboidal HLCs were observed,and cells also expressed hepatocyte-specific marker genes including albumin(ALB),α-fetoprotein(AFP),cytokeratin 18/19(CK18/19),and cytochrome P450 1A1/3A4(CYP1A1/3A4).Differentiated cells further demonstrated in vitro mature hepatocyte functions such as urea synthesis,glycogen storage,and indocyanine green(ICG)uptake.After intrasplenic transplantation into mice with 2/3 partial hepatectomy,the MenSC-derived HLCs were detected in recipient livers and expressed human ALB protein.We also showed that MenSC-derived HLC transplantation could restore the serum ALB level and significantly suppressed transaminase activity of liver injury animals.In conclusion,MenSCs may serve as an ideal,easily accessible source of material for tissue engineering and cell therapy of liver tissues.
文摘In this work,stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated.Some effective sufficient conditions of stability,uniform stability,asymptotic uniform stability and Mittag Leffler stability.The approach presented is based on the specially introduced piecewise continuous Lyapunov functions.Furthermore,some numerical examples are given to show the effectiveness of our obtained theoretical results.
基金partially supported by the NSFC(11271227,11271161)the PCSIRT(IRT1264)the Fundamental Research Funds of Shandong University(2017JC019)。
文摘In this paper,we mainly investigate the value distribution of meromorphic functions in Cmwith its partial differential and uniqueness problem on meromorphic functions in Cmand with its k-th total derivative sharing small functions.As an application of the value distribution result,we study the defect relation of a nonconstant solution to the partial differential equation.In particular,we give a connection between the Picard type theorem of Milliox-Hayman and the characterization of entire solutions of a partial differential equation.
文摘In this paper, we shall be concerned with the numerical solution of parabolic equations in one space variable and the time variable t. We expand Taylor series to derive a higher-order approximation for U<sub>t</sub>. We begin with the simplest model problem, for heat conduction in a uniform medium. For this model problem, an explicit difference method is very straightforward in use, and the analysis of its error is easily accomplished by the use of a maximum principle. As we shall show, however, the numerical solution becomes unstable unless the time step is severely restricted, so we shall go on to consider other, more elaborate, numerical methods which can avoid such a restriction. The additional complication in the numerical calculation is more than offset by the smaller number of time steps needed. We then extend the methods to problems with more general boundary conditions, then to more general linear parabolic equations. Finally, we shall discuss the more difficult problem of the solution of nonlinear equations.
