A Study Of BVP4C Method For Solving Boundary.pptx

Bvp4c and odes presentation

1. A Study o f BVP4C Method for Solving Boundary Value Problems Department of Mathematics

2. Project Outlines Introduction Description of BVP4C Method Solution of Boundary Value Problems using BVP4C Beam Deflection Problems

3. I ntroduction Differential Equation O rder o f a Differential E quation D egree of the Differential E quation Solution of First Order Differential Equations Separable Equations Homogeneous Equations Exact Equations Linear Equations Initial and Boundary V alue P roblems

4. Introduction Differential Equation: An equation containing the derivatives or differentials of one or more dependent variables with respect to one or more independent variables is said to be a differential equation (DE ). O rder and Degree of a Differential E quation: By order of differential equation we mean t he order of the highest derivative present in the equation and t he degree of the differential equation is represented by the power of the highest order derivative in the given differential equation : The order of differentia equation is 4 and the exponent of the highest order derivative is one. Hence , the degree of this equation is one . Solution Of First Order Differential Equations : There are few methods for solving first order Differential Equations.  

5. A differential equation of form is said to be separable or to have separable variables. The equation P(x, y)dx + Q(x, y ) dy = is an exact differential equation if there exists a function u of two variables x and y having continuous partial derivatives such that the exact differential equation definition is separated as follows: u x (x , y) = P(x, y ) and u y (x, y) = Q(x, y ), Therefore , the general solution of the equation is u(x, y) = C . If a function x has a property that f( tx,ty ) = t n f ( x,y ) for some real number n, then f is said to be homogeneous function of degree n. A differential equation of form a 1 (x) is said to be linear equation [3]. An Initial V alue P roblem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. A B oundary V alue Problem (BVP) is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.  

6. Description of BVP4C Method BVP4C Description Odefun Bcfun Solinit Xmesh Y init-initial guess of the solution

7. Description of BVP4C Method For Solving BVPs BVP4C: Solve boundary value problem — fourth-order method . BVP4C is a finite difference method code that implement 3 step Lobatto formula. It provides continuous solutions that are fourth order accurate uniformly in the interval of integration . Description: sol = bvp4c( odefun,bcfun,solinit ) integrates a system of differential equations of the form y’ = f( x,y ) specified by odefun, subject to the boundary conditions described by bcfun and the initial solution guess solinit . Odefun: Functions to solve, specified as a function handle that defines the functions to be integrated . O defun and bcfun must accept the same number of input arguments. To code odefun , we generally use the functional signature dydx = odefun( x,y ). Bcfun: To code bcfun, use the functional signature res = bcfun( ya,yb ). For example, if y(a) = 1 and y(b) = 0, then the residual value of ya(1)-1 should be 0 at the point x = a . Similarly , since y(b) = 0, the residual value of yb (1) should be 0 at the point x =b.

8. Solinit: solinit = bvpinit ( x,yinit ) uses the initial mesh x and initial solution guess yinit to form an initial guess of the solution for a boundary value problem. You then can use the initial guess solinit as one of the inputs to bvp4c to solve the boundary value problem. Xmesh : Initial mesh, specified as a vector. To solve the problem on the interval [a , b ] , specify “ x(1)” as a and “x(end)” as b . Y init-initial guess of the solution: Forming a good initial guess of the solution to a BVP problem is perhaps the most difficult part of solving the problem. BVP solutions are not necessarily unique, so the initial guess can be the deciding factor in which of many solutions the solver returns . Creating a good initial guess for the solution is more an art than a science. However, some general guidelines include: Have the initial guess satisfy the boundary conditions, since the solution is required to satisfy them as well. Consider the placement of the mesh points (the x-coordinates of the initial guess of the solution).

9. Solution of Boundary Value Problems using BVP4C Second Order Non-Linear Boundary Value Problem Applying BVP4C MATLAB Code for BVP BVP Using Different Initial Guess MATLAB Code for BVP Result Using Different Initial Guess

10. Solution of Boundary Value Problems U sing BVP4C Second Order Non-Linear Boundary Value Problem [2] (1) together with Boundary Conditions y(0) = y(1 ) = . Applying BVP4C converting equation(1) into two first order ode’s: Putting y(1 ) = y, y(2) = y′ so we have; y ′(1) = y′, y′(2) = y ′′, Hence, y′(1) = y(2) and y ′(2) = The new boundary conditions are: y(0) = 0 at x = and y(1 ) = at x = 1 .  

11. MATLAB Code for BVP

12. BVP U sing Different I nitial G uess Changing initial guess by [ 3 ,0]

13. Result Using Different Initial Guess

14. Beam Deflection The Beam is a long piece of a body that is capable to hold the load by resisting the bending . The deflection of the beam towards in a particular direction when force is applied to it is known as Beam deflection. (2) where y is the deflection. The given equation under the appropriate boundary conditions determines the deflection of a loaded beam. Classification of Beam There are two types of beam classification: Cantilever Beam has got fixed support at one end while the other end is free to move . Simply Supported Beam has got support at both ends . The governing differential equation is: together with Boundary Conditions , y(0) = y(L) = y ′′(0) = y′′(L) = 0.  

15. Applying BVP4C: converting equation into four first order ode’s by putting: y(1) = y and y(2) = y′ , so we get y′(1) = y(2) and y′(2) = y(3), also we have y′(3) =y(4) and y′(4) =q/EL. The new boundary conditions are: y(1) = 0 at x = 0 , y(1) = 0 at x = L , y(3) = 0 at x = 0 , and y(3 ) = 0 at x = L.

16. Matlab code for BVP

17. References [1] Shampine , L., Gladwell, I., and Thompson, S. (2003). Solving ODEs with MATLAB. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511615542 [2] Shampine, L.F., M.W. Reichelt, and J. Kierzenka. "Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c." MATLAB File Exchange, 2004. [3] Zill, Dennis G. A First Course in  Differential Equations  with Modeling Applications. Cengage Learning,2012.