Week 2.13 - Literature review of Boundary and Initial Conditions
Week 2.13 - Literature review of Boundary and Initial Conditions
Aim : Providing an explanation of Initial and Boundary Conditions, with types of Partial Differential Equations.
Whenever a differential equation is solved, it may have many solutions, for case of the general solution having arbitary constants, say c1 adn c2. "Initial conditions" are those which are defined at some specific point, for instance at 4, which coulc be like y(4)=8. This implies that, y being a function of x, at x =4, the value of y will be 8. Taking this initial condition into consideration, and applying to the general solution of a differential equation, gives values of constants c1 and c2 and one particular solution. Initial conditions corresponds to, say some values, at the beginning of a analysis or simulation.
On the other hand, assume that the conditions are defined for not only one point, but two, i.e. y(4)=8 and y(5) = 9, such conditions might give a unique solution, many or no solution. These "Boundary conditions", like constraints, are defined for any two points in an interval, the second point could be an end point too. The solution to an initial condition yields a definite solution whereas, with Boundary conditions result in different cases.
Suppose an analysis has to be done, a simulation of pipe flow, here the initial conditions could be assigned as inlet fluid velocity and pressure are 'a m/s' and 'b bar', however at the downstream section, the velocity may not be known, so here the boundary condition which can be applied could be 'c bar' (as atmospheric pressure). The flow velocity in a closed tube, while flowing will get reduced because of wall friction, and at the outlet, the velocity magnitude obtained will correspond to that of the applied boundary condition of pressure.
- NEED FOR BOUNDARY CONDITION
Defining an appropriate boundary condition is necessay in solving an equation, if not, then it may not let the solution to be converge, giving a wrong output of a predescribed problem.
Mainly, but there could be more, Dirichlet, Neumann, Cauchy and Robin boundary conditions are used for solving equations.
- TYPES OF PARTIAL DIFFERENTIAL EQUATIONS
1. First order linear PDE - Equation has just 1 variable, first order derivative.
`(delf)/(delx)+(delf)/(dely)=0`
2. First order non-linear PDE - Equation is either squared or multiplied with a variable
`u(delf)/(delx)+(delf)/(dely)=0` or `(delf)/(delx)+((delf)/(dely))^2=0`
3. Second order linear PDE - Equation is of 2nd order derivative, can be w.r.t. x, y or x and y both.
`(del^2f)/(delx^2)+ (del^2f)/(delxdely)+(del^2f)/(dely^2)=0`
3. Second order non-linear PDE - Equation is of 2nd order derivative and multiplied with a variable, can be w.r.t. x, y or x and y both.
`u((∂^2f)/(∂x^2))+(∂^2f)/(∂y^2)=0`
4. Homogeneous PDE - The R.H.S. of the equation is zero. This includes both 1st order and 2nd order equation.
`(∂^2f)/(∂x^2)+(∂^2f)/(∂y^2)=0` or `(∂f)/(∂x)+(∂f)/(∂y)=0`
5. Non-Homogeneous PDE - The R.H.S. of the equation is not zero, has some function. This includes both 1st order and 2nd order equation.
`(del^2f)/(delx^2)+(delf)/(dely)=fsinx`