With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a res...With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a result, their abundant new soliton-like solutions and period form solutions are found.展开更多
The method of Riccati equation is extended for constructing travelling wave solutions of nonlinear partial differential equations. It is applied to solve the Karamoto-Sivashinsky equation and then its more new explici...The method of Riccati equation is extended for constructing travelling wave solutions of nonlinear partial differential equations. It is applied to solve the Karamoto-Sivashinsky equation and then its more new explicit solutions have been obtained. From the results given in this paper, one can see the computer algebra plays an important role in this procedure.展开更多
For the purpose of overcoming the difficulty of the so-called 'intermediate expression swell' in applying computer algebra, a semi-inverse algorithm is proposed. The or del of seeking solutions for various pro...For the purpose of overcoming the difficulty of the so-called 'intermediate expression swell' in applying computer algebra, a semi-inverse algorithm is proposed. The or del of seeking solutions for various problems is partly inverted, i. e., the intermediate expressions appearing in computation are 'frozen' in the symbolic form at first, and 'unfrozen' till the formal expressions of final solutions are found out. In this rca!, the overflow due to the shortage of saving space is avoided. The applications of the algorithm in the problems on nonlinear oscillation, dynamical optimization and interfacial solitary waves are described, which show the effectiveness of the semi-inverse algorithm.展开更多
Computer Algebra Systems have been extensively used in higher education. The reasons are many e.g., visualize mathematical problems, correlate real-world problems on a conceptual level, are flexible, simple to use, ac...Computer Algebra Systems have been extensively used in higher education. The reasons are many e.g., visualize mathematical problems, correlate real-world problems on a conceptual level, are flexible, simple to use, accessible from anywhere, etc. However, there is still room for improvement. Computer algebra system (CAS) optimization is the set of best practices and techniques to keep the CAS running optimally. Best practices are related to how to carry out a mathematical task or configure your system. In this paper, we are going to examine these techniques. The documentation sheets of CASs are the source of data that we used to compare them and examine their characteristics. The research results reveal that there are many tips that we can follow to accelerate performance.展开更多
This report shows how starting from classic electric circuits embodying commonly electric components we have reached semi-complicated circuits embodying the same components that analyzing the signal characteristics re...This report shows how starting from classic electric circuits embodying commonly electric components we have reached semi-complicated circuits embodying the same components that analyzing the signal characteristics requires a Computer Algebra System. Our approach distinguishes itself from the electrical engineers’ (EE) approach that relies on utilizing commercially available software. Our approach step-by-step shows how Kirchhoff’s rules are applied conducive to the needed circuit information. It is shown for the case at hand the characteristic information is a set of coupled differential equations and that with the help of Mathematica numeric solutions are sought. Our report paves the research road for unlimited creative similar circuits with any degree of complications. Occasionally, by tweaking the circuits we have addressed the “what if” scenarios widening the scope of the investigation. Justification of the accuracy of our analysis for the generalized circuits is cross-checked by arranging the components symmetrizing the circuit leading to an intuitively predictable reasonable result. Mathematica codes are embedded assisting the interested reader in producing and extending our results.展开更多
The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations . T...The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations . The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales .展开更多
<span style="font-size:10.0pt;font-family:;" "="">Nucleons are fermions with intrinsic spins exhibiting dipole character. Dipole-dipole interaction via their dipole moments is the key fe...<span style="font-size:10.0pt;font-family:;" "="">Nucleons are fermions with intrinsic spins exhibiting dipole character. Dipole-dipole interaction via their dipole moments is the key feature quantifying the short-range nucleonics interaction in two-body physics. For a pair of interacting dipoles, the energy of a pair is the quantity of interest. The same is true for chemical polar molecules. For both cases, derivation of energy almost exclusively is carried out vectorially </span><span style="font-size:10.0pt;font-family:" color:#943634;"=""><a href="#ref1">[1]</a></span><span style="font-size:10.0pt;font-family:;" "=""></span><span style="font-size:10.0pt;font-family:" times="" new="" roman",serif;"=""><span></span></span><span style="font-size:10.0pt;font-family:;" "="">. Although uncommon the interacting energy can be derived algebraically too. For the latter Taylor, expansion is applied </span><span style="font-size:10.0pt;font-family:" color:#943634;"=""><a href="#ref2">[2]</a></span><span style="font-size:10.0pt;font-family:" background:lime;"=""></span><span style="font-size:10.0pt;font-family:" minion="" pro="" capt",serif;background:lime;"=""><span></span></span><span style="font-size:10.0pt;font-family:;" "="">.</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">The given expression although appears to be correct it is incomplete.</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">In our report,</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">by applying Taylor</span><span style="font-size:10.0pt;font-family:;" "="">’</span><span style="font-size:10.0pt;font-family:;" "="">s expansion up to the 4th order and utilizing a Computer Algebra System we formulate the missing terms.</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">Our report highlights the impact of correcting missing terms by giving two explicit examples.</span><span style="font-family:;" "=""></span>展开更多
A new algorithm for symbolic computation of polynomial-type conserved densities for nonlinear evolution systems is presented. The algorithm is implemented in Maple. The improved algorithm is more efficient not only in...A new algorithm for symbolic computation of polynomial-type conserved densities for nonlinear evolution systems is presented. The algorithm is implemented in Maple. The improved algorithm is more efficient not only in removing the redundant terms of the genera/form of the conserved densities but also in solving the conserved densities with the associated flux synchronously without using Euler operator. Furthermore, the program conslaw.mpl can be used to determine the preferences for a given parameterized nonlinear evolution systems. The code is tested on several well-known nonlinear evolution equations from the soliton theory.展开更多
In this article, based on the Taylor expansions of generating functions and stepwise refinement procedure, authors suggest a algorithm for finding the Lie and high (generalized) symmetries of partial differential equa...In this article, based on the Taylor expansions of generating functions and stepwise refinement procedure, authors suggest a algorithm for finding the Lie and high (generalized) symmetries of partial differential equations (PDEs). This algorithm transforms the problem having to solve over-determining PDEs commonly encountered and difficulty part in standard methods into one solving to algebraic equations to which one easy obtain solution. so, it reduces significantly the difficulties of the problem and raise computing efficiency. The whole procedure of the algorithm is carried out automatically by using any computer algebra system. In general, this algorithm can yields many more important symmetries for PDEs.展开更多
It is a common misconception that electric “resistance” always is a positive defined electric element. <em>i.e.</em>, the plot of the voltage across the resistor, V vs. its current, i is a slanted straig...It is a common misconception that electric “resistance” always is a positive defined electric element. <em>i.e.</em>, the plot of the voltage across the resistor, V vs. its current, i is a slanted straight line with a positive slope. Esaki diode also known as tunnel diode is an exception to this character. For a certain voltage range, the current recedes resulting in a line with a negative slope;it is interpreted as negative resistance. In this research flavored report, we investigate the impact of the negative resistance in a typical classic electric circuit. E.g., a tunnel diode, D is inserted in a classic electric circuit that is composed of an ohmic resistor, R and a capacitor, C which are all in series with a DC power supply. The circuit equation for the RCD circuit is a nonlinear ordinary differential equation (NLODE). In line with the ever-growing popular Computer Algebra System (CAS), this is solved numerically utilizing two distinctly different CASs. The consistency of the solutions confidently leads to the understanding of the impact of the negative resistance. The circuit characteristics are compared to the classic analogous RC circuit. The report embodies an atlas of characteristics of the circuits making the analysis visually comprehensible.展开更多
基金The project supported by the National Key Basic Research Development Project Program under Grant No.G1998030600the Foundation of Liaoning Normal University
文摘With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a result, their abundant new soliton-like solutions and period form solutions are found.
文摘The method of Riccati equation is extended for constructing travelling wave solutions of nonlinear partial differential equations. It is applied to solve the Karamoto-Sivashinsky equation and then its more new explicit solutions have been obtained. From the results given in this paper, one can see the computer algebra plays an important role in this procedure.
文摘For the purpose of overcoming the difficulty of the so-called 'intermediate expression swell' in applying computer algebra, a semi-inverse algorithm is proposed. The or del of seeking solutions for various problems is partly inverted, i. e., the intermediate expressions appearing in computation are 'frozen' in the symbolic form at first, and 'unfrozen' till the formal expressions of final solutions are found out. In this rca!, the overflow due to the shortage of saving space is avoided. The applications of the algorithm in the problems on nonlinear oscillation, dynamical optimization and interfacial solitary waves are described, which show the effectiveness of the semi-inverse algorithm.
文摘Computer Algebra Systems have been extensively used in higher education. The reasons are many e.g., visualize mathematical problems, correlate real-world problems on a conceptual level, are flexible, simple to use, accessible from anywhere, etc. However, there is still room for improvement. Computer algebra system (CAS) optimization is the set of best practices and techniques to keep the CAS running optimally. Best practices are related to how to carry out a mathematical task or configure your system. In this paper, we are going to examine these techniques. The documentation sheets of CASs are the source of data that we used to compare them and examine their characteristics. The research results reveal that there are many tips that we can follow to accelerate performance.
文摘This report shows how starting from classic electric circuits embodying commonly electric components we have reached semi-complicated circuits embodying the same components that analyzing the signal characteristics requires a Computer Algebra System. Our approach distinguishes itself from the electrical engineers’ (EE) approach that relies on utilizing commercially available software. Our approach step-by-step shows how Kirchhoff’s rules are applied conducive to the needed circuit information. It is shown for the case at hand the characteristic information is a set of coupled differential equations and that with the help of Mathematica numeric solutions are sought. Our report paves the research road for unlimited creative similar circuits with any degree of complications. Occasionally, by tweaking the circuits we have addressed the “what if” scenarios widening the scope of the investigation. Justification of the accuracy of our analysis for the generalized circuits is cross-checked by arranging the components symmetrizing the circuit leading to an intuitively predictable reasonable result. Mathematica codes are embedded assisting the interested reader in producing and extending our results.
文摘The method of boundary layer with multiple scales and computer algebra were applied to study the asymptotic behavior of solution of boundary value problems for a class of system of nonlinear differential equations . The asymptotic expansions of solution were constructed. The remainders were estimated. And an example was analysed. It provides a new foreground for the application of the method of boundary layer with multiple scales .
文摘<span style="font-size:10.0pt;font-family:;" "="">Nucleons are fermions with intrinsic spins exhibiting dipole character. Dipole-dipole interaction via their dipole moments is the key feature quantifying the short-range nucleonics interaction in two-body physics. For a pair of interacting dipoles, the energy of a pair is the quantity of interest. The same is true for chemical polar molecules. For both cases, derivation of energy almost exclusively is carried out vectorially </span><span style="font-size:10.0pt;font-family:" color:#943634;"=""><a href="#ref1">[1]</a></span><span style="font-size:10.0pt;font-family:;" "=""></span><span style="font-size:10.0pt;font-family:" times="" new="" roman",serif;"=""><span></span></span><span style="font-size:10.0pt;font-family:;" "="">. Although uncommon the interacting energy can be derived algebraically too. For the latter Taylor, expansion is applied </span><span style="font-size:10.0pt;font-family:" color:#943634;"=""><a href="#ref2">[2]</a></span><span style="font-size:10.0pt;font-family:" background:lime;"=""></span><span style="font-size:10.0pt;font-family:" minion="" pro="" capt",serif;background:lime;"=""><span></span></span><span style="font-size:10.0pt;font-family:;" "="">.</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">The given expression although appears to be correct it is incomplete.</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">In our report,</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">by applying Taylor</span><span style="font-size:10.0pt;font-family:;" "="">’</span><span style="font-size:10.0pt;font-family:;" "="">s expansion up to the 4th order and utilizing a Computer Algebra System we formulate the missing terms.</span><span style="font-size:10.0pt;font-family:;" "=""> </span><span style="font-size:10.0pt;font-family:;" "="">Our report highlights the impact of correcting missing terms by giving two explicit examples.</span><span style="font-family:;" "=""></span>
文摘A new algorithm for symbolic computation of polynomial-type conserved densities for nonlinear evolution systems is presented. The algorithm is implemented in Maple. The improved algorithm is more efficient not only in removing the redundant terms of the genera/form of the conserved densities but also in solving the conserved densities with the associated flux synchronously without using Euler operator. Furthermore, the program conslaw.mpl can be used to determine the preferences for a given parameterized nonlinear evolution systems. The code is tested on several well-known nonlinear evolution equations from the soliton theory.
文摘In this article, based on the Taylor expansions of generating functions and stepwise refinement procedure, authors suggest a algorithm for finding the Lie and high (generalized) symmetries of partial differential equations (PDEs). This algorithm transforms the problem having to solve over-determining PDEs commonly encountered and difficulty part in standard methods into one solving to algebraic equations to which one easy obtain solution. so, it reduces significantly the difficulties of the problem and raise computing efficiency. The whole procedure of the algorithm is carried out automatically by using any computer algebra system. In general, this algorithm can yields many more important symmetries for PDEs.
文摘It is a common misconception that electric “resistance” always is a positive defined electric element. <em>i.e.</em>, the plot of the voltage across the resistor, V vs. its current, i is a slanted straight line with a positive slope. Esaki diode also known as tunnel diode is an exception to this character. For a certain voltage range, the current recedes resulting in a line with a negative slope;it is interpreted as negative resistance. In this research flavored report, we investigate the impact of the negative resistance in a typical classic electric circuit. E.g., a tunnel diode, D is inserted in a classic electric circuit that is composed of an ohmic resistor, R and a capacitor, C which are all in series with a DC power supply. The circuit equation for the RCD circuit is a nonlinear ordinary differential equation (NLODE). In line with the ever-growing popular Computer Algebra System (CAS), this is solved numerically utilizing two distinctly different CASs. The consistency of the solutions confidently leads to the understanding of the impact of the negative resistance. The circuit characteristics are compared to the classic analogous RC circuit. The report embodies an atlas of characteristics of the circuits making the analysis visually comprehensible.
