Numerical Approximation Of Boundary Value Problem Of A Class Of Third-order Ordinary Differential Equation From Draining And Coating Flows

In this paper we consider the numerical computational methods for a class of boundary value problems from draining and coating flows involving a third-order ordinary differential equation.wheresatisfies the following hypothesis:(H1) f(y) = (1 - y)~λg(y), whereλ> 0, g(y) is continuous and nonincreasing on (0,1], and g(1)≥1.There is an existence theorem of the solution to the equation(1), (2).Theorem 1 [6] Assume that f(y) satisfies the hypothesis (H1). The boundary value problems (1), (2) has a solutionMoreover, ifλ> 1, thenifλ= 1, thenLater on, the hypothesis(H1) is weaken to(H2)/(y) = (1 - y)~λg(y), whereλ> 0, g(y) is a positive continuous function defined on (0,1], and there exists a function G(y)∈C(0,1], which is nonincreasing in y such thatTheorem 2 [7] Assume that f(y) satisfies the hypothesis (H2). The boundary value problems (1), (2) has a unique solution.In order to solve the third-order equation, we first consider following initial value problem of the formTheorem 3 [6] Suppose that f(t) satisfies (H1). Then the initial value problem (3), (4) has a solution and satisfiesTheorem 4 [7] Suppose that f(t) satisfies (H2). Then the initial value problem (3), (4) has a unique solutionA numerical methods for solving such problems is studied in [8]. In this paper we set up two numerical methods for solving the third-order problem (1), (2). We letwhere 0 = t1 < t2 < t3< ... < tn = 1 and w(s) is the numerical solution of the equation (3), (4). The inverse function of u(t) is the solution of the equation (1), (2). Another method is to translate the infinite boundary condition into finite boundary condition, i.e. then we use the solution of this equation with large M to approximate the solution of (1), (2).In the end, we apply these two methods to deal with examples and discuss the error estimate of the numerical methods.