Introduction Algorithms and Theorems for approximating solutions of two-point boundary value problems An Algorithm for Approximating Solutions On the Long Intervals Volterra Equation Using Auxiliary Variables According To Parker-Sochacki Modified Picard Iteration Applied to Boundary Value Problems and Volterra Integral Equations. Mathematical Association of America MD-DC-VA Section, November 7 & 8, 2014 Bowie State University Bowie, Maryland Hamid Semiyari November 7, 2014 Introduction Algorithms and Theorems for approximating solutions of two-point boundary value problems An Algorithm for Approximating Solutions On the Long Intervals Volterra Equation Using Auxiliary Variables According To Parker-Sochacki TABLE OF CONTENTS Introduction Algorithms and Theorems for approximating solutions of two-point boundary value problems An Algorithm for Approximating Solutions On the Long Intervals Volterra Equation Using Auxiliary Variables According To Parker-Sochacki Introduction Algorithms and Theorems for approximating solutions of two-point boundary value problems An Algorithm for Approximating Solutions On the Long Intervals Volterra Equation Using Auxiliary Variables According To Parker-Sochacki INTRODUCTION In this talk we present a new algorithm for approximating solutions of two-point boundary value problems and provide a theorem that gives conditions under which it is guaranteed to succeed. We show how to make the algorithm computationally efficient and demonstrate how the full method works both when guaranteed to do so and more broadly. In the first section the original idea and its application are presented. We also show how to modify the same basic idea and procedure in order to apply it to problems with boundary conditions of mixed type. In the second section we introduce a new algorithm for the case if the original algorithm failed to converge on a long interval. We split the long interval into subintervals and show the new algorithm gives convergence to the solution. Finally, we repose a Volterra equation using auxiliary variables according to Parker-Sochacki in such a way that the solution can be approximated by the Modified Picard iteration scheme. Introduction Algorithms and Theorems for approximating solutions of two-point boundary value problems An Algorithm for Approximating Solutions On the Long Intervals Volterra Equation Using Auxiliary Variables According To Parker-Sochacki The algorithm that will be developed in here for the solution of certain boundary value problems is based on a kind of Picard iteration as just described. It is well known that in many instances Picard iteration performs poorly in actual practice, even for relatively simple differential equations. A great improvement can sometimes be had by exploiting ideas originated by G. Parker and J. Sochacki [7]. The Parker-Sochacki method (PSM) is an algorithm for solving systems of ordinary differential equations (ODEs). The method produces Maclaurin series solutions to systems of differential equations, with the coefficients in either algebraic or numerical form. The Parker-Sochacki method convert the initial value problem to a system that the right hand side is polynomials, consequently the right hand side then are continuous and satisfy the Lipschitz condition on an open region containing the initial point t = 0. Moreover, the system guarantees that the iterative integral is easy to integrate. We begin with the following example, the initial value problem y0 = siny; y(0) = p=2: (1) Introduction Algorithms and Theorems for approximating solutions of two-point boundary value problems An Algorithm for Approximating Solutions On the Long Intervals Volterra Equation Using Auxiliary Variables According To Parker-Sochacki To approximate it by means of Picard iteration, from the integral equation that is equivalent to (1), namely, Z t y(t) = p + siny(s)ds 2 0 we define the Picard iterates Z t p p y0(t) ≡ ; yk+1(t) = + sinyk(s)ds for k ≥ 0: 2 2 0 The first few iterates are p y0(t) = 2 p y1(t) = 2 + t p p y2(t) = 2 − cos( 2 + t) Z t p y3(t) = + cos(sins)ds; 2 0 the last of which cannot be expressed in closed form. We define variables u and v by u = siny and v = cosy so that y0 = u, u0 = (cosy)y0 = uv, and v0 = (−siny)y0 = −u2, hence the original problem is embedded as the first component in the three-dimensional problem y0 = u p y(0) = 2 0 u = uv u(0) = 1 v0 = −u2 v(0) = 0 Although the dimension has increased, now the right hand sides are all polynomial functions so quadratures can be done easily. Introduction Algorithms and Theorems for approximating solutions of two-point boundary value problems An Algorithm for Approximating Solutions On the Long Intervals Volterra Equation Using Auxiliary Variables According To Parker-Sochacki In particular, the Picard iterates are now 0 p 1 0 p 1 0 u (s) 1 2 Z t 2 Z t k y0(t) = @ 1 A yk+1(t) = y0 + yk(s)ds = @ 1 A + @uk(s)vk(s)A ds: 0 0 2 0 0 −uk (s) The first component of the first few iterates are p y0(t) = 2 p y1(t) = 2 + t p y2(t) = 2 + t p 1 3 y3(t) = 2 + t − 6 t p 1 3 1 5 y4(t) = 2 + t − 6 t + 24 t p 1 3 1 5 y5(t) = 2 + t − 6 t + 24 t p 1 3 1 5 61 7 y6(t) = 2 + t − 6 t + 24 t − 5040 t The exact solution to the problem (1) is y(t) = 2arctan(et) the first nine terms of its Maclaurin series are p 1 3 1 5 61 7 9 2 + t − 6 t + 24 t − 5040 t + O(t ) As we can see the system of ODE proposed by Parker and Sochacki is generating a Maclaurin series solution for problem (1). Introduction Algorithms and Theorems for approximating solutions of two-point boundary value problems An Algorithm for Approximating Solutions On the Long Intervals Volterra Equation Using Auxiliary Variables According To Parker-Sochacki AN EFFICIENT ALGORITHM FOR APPROXIMATING SOLUTIONS OF TWO-POINT BOUNDARY VALUE PROBLEMS Consider a two-point boundary value problem of the form y00 = f (t;y;y0); y(a) = a; y(b) = b: (2) If f is locally Lipschitz in the last two variables then by the Picard-Lindelof¨ Theorem, for any g 2 R the initial value problem y00 = f (t;y;y0); y(a) = a; y0(a) = g (3) will have a unique solution on some interval about t = a. Introducing the variable u = y0 we obtain the first order system that is equivalent to (3), y0 = u u0 = f (t;y;u) (4) y(a) = a; u(a) = g Introduction Algorithms and Theorems for approximating solutions of two-point boundary value problems An Algorithm for Approximating Solutions On the Long Intervals Volterra Equation Using Auxiliary Variables According To Parker-Sochacki The equivalent integral equation to (3) will be Z t y(t) = a + g(t − a) + (t − s)f (s;y(s);y0(s))ds; (5) a which we then solve, when evaluated at t = b, for g: Z b g = 1 b − a − (b − s)f (s;y(s);y0(s))ds (6) b−a a The key idea is we use picard iteration to obtain successive approximations to the value of g. Thus the iterates are y[0](t) ≡ a [0] b−a u (t) ≡ b−a (7a) [0] b−a g ≡ b−a and Z t y[k+1](t) = a + u[k](s)ds a Z t u[k+1](t) = g[k] + f (s;y[k](s);u[k](s))ds (7b) a 1 Z b g[k+1] = b − a − (b − s)f (s;y[k](s);u[k](s))ds : b − a a Introduction Algorithms and Theorems for approximating solutions of two-point boundary value problems An Algorithm for Approximating Solutions On the Long Intervals Volterra Equation Using Auxiliary Variables According To Parker-Sochacki THEOREM Theorem Let f : [a;b] × R2 ! R : (t;y;u) 7! f (t;y;u) be Lipschitz in y = (y;u) with Lipschitz constant L with respect to absolute value on R and the 2 3 −1 sum norm on R . If 0 < b − a < (1 + 2 L) then for any a, b 2 R the boundary value problem y00 = f (t;y;y0); y(a) = a; y(b) = b (8) has a unique solution. Introduction Algorithms and Theorems for approximating solutions of two-point boundary value problems An Algorithm for Approximating Solutions On the Long Intervals Volterra Equation Using Auxiliary Variables According To Parker-Sochacki EXAMPLES In this example we treat a problem in which the function f on the right hand side fails to be Lipschitz yet Algorithm nevertheless performs well. Consider the boundary value problem y00 = −e−2y; y(0) = 0; y(1:2) = lncos1:2 ≈ −1:015123283; (9) for which the unique solution is y(t) = lncost, yielding g = 0. Introducing the dependent variable u = y0 to obtain the equivalent first order system y0 = u, u0 = −e−2y and the variable v = e−2y to replace the transcendental function with a polynomial we obtain the expanded system y0 = u u0 = −v v0 = −2uv Introduction Algorithms and Theorems for approximating solutions of two-point boundary value problems An Algorithm for Approximating Solutions On the Long Intervals Volterra Equation Using Auxiliary Variables According To Parker-Sochacki y[0](t) ≡ 0 lncos1:2 u[0](t) ≡ 1:2 (10) v[0](t) ≡ 1 lncos1:2 g[0] = 1:2 and 0 1 y[k+1](t) 0 0 1 B [k+1] C = 1 R 1:2 [k] (11) @u (t)A @ 1:2 (b − a + 0 (1:2 − s)v (s)dsA v[k+1](t) 1 0 [k] 1 Z t u (s) + B −v[k](s) C ds; 0 @ A −2u[k](s)v[k](s) where we have shifted the update of g and incorporated it into the update of u[k](t).