Nearly all scientists,at conjunction with simplifying a differential equation,have probably used dimensional analysis.Dimensional analysis(also called the Factor-Label Method or the Unit Factor Method)is an approach t...Nearly all scientists,at conjunction with simplifying a differential equation,have probably used dimensional analysis.Dimensional analysis(also called the Factor-Label Method or the Unit Factor Method)is an approach to the problem that uses the fact that one can multiply any number or expression without changing its value.This is a useful technique.However,the reader should take care to understand that chemistry is not simply a mathematics problem.In every physical problem,the result must match the real world.In physics and science,dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions.The dimension of a physical quantity is the combination of the fundamental physical dimensions(usually mass,length,time,electric charge,and temperature)which describe it;for example,speed has the dimension length/time,and may be measured in meters per second,miles per hour,or other units.Dimensional analysis is necessary because a physical law must be independent of the units used to measure the physical variables in order to be general for all cases.One of the most derivation elements from dimensional analysis is scaling and consequently arriving at similarity methods that branch out to two different groups namely self-similarity as the first one,and second kind that through them one can solve the most complex none-linear ODEs(Ordinary Differential Equations)and PDEs(Partial Differential Equations)as well.These equations can be solved either in Eulearian or Lagrangian coordinate systems with their associated BCs(Boundary Conditions)or ICs(Initial Conditions).Exemplary ODEs and PDEs in the form of none-linear can be seen in strong explosives or implosives scenario,where the results can easily be converted to induction of energy in a control forms for a peaceful purpose(i.e.,fission or fusion reactions).展开更多
文摘Nearly all scientists,at conjunction with simplifying a differential equation,have probably used dimensional analysis.Dimensional analysis(also called the Factor-Label Method or the Unit Factor Method)is an approach to the problem that uses the fact that one can multiply any number or expression without changing its value.This is a useful technique.However,the reader should take care to understand that chemistry is not simply a mathematics problem.In every physical problem,the result must match the real world.In physics and science,dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions.The dimension of a physical quantity is the combination of the fundamental physical dimensions(usually mass,length,time,electric charge,and temperature)which describe it;for example,speed has the dimension length/time,and may be measured in meters per second,miles per hour,or other units.Dimensional analysis is necessary because a physical law must be independent of the units used to measure the physical variables in order to be general for all cases.One of the most derivation elements from dimensional analysis is scaling and consequently arriving at similarity methods that branch out to two different groups namely self-similarity as the first one,and second kind that through them one can solve the most complex none-linear ODEs(Ordinary Differential Equations)and PDEs(Partial Differential Equations)as well.These equations can be solved either in Eulearian or Lagrangian coordinate systems with their associated BCs(Boundary Conditions)or ICs(Initial Conditions).Exemplary ODEs and PDEs in the form of none-linear can be seen in strong explosives or implosives scenario,where the results can easily be converted to induction of energy in a control forms for a peaceful purpose(i.e.,fission or fusion reactions).
