Modified Picard Iteration Applied to Boundary Value Problems And

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

TABLEOFCONTENTS

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) = π/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) = π + siny(s)ds 2 0

we deﬁne the Picard iterates Z t π π y0(t) ≡ , yk+1(t) = + sinyk(s)ds for k ≥ 0. 2 2 0 The ﬁrst few iterates are

π y0(t) = 2 π y1(t) = 2 + t π π y2(t) = 2 − cos( 2 + t) Z t π 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 π 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

 π   π   u (s)  2 Z t 2 Z t k y0(t) =  1  yk+1(t) = y0 + yk(s)ds =  1  + uk(s)vk(s) ds. 0 0 2 0 0 −uk (s)

The ﬁrst component of the ﬁrst few iterates are

π y0(t) = 2 π y1(t) = 2 + t π y2(t) = 2 + t π 1 3 y3(t) = 2 + t − 6 t π 1 3 1 5 y4(t) = 2 + t − 6 t + 24 t π 1 3 1 5 y5(t) = 2 + t − 6 t + 24 t π 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

π 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 ALGORITHMFOR APPROXIMATING SOLUTIONSOF TWO-POINT BOUNDARY VALUE PROBLEMS

Consider a two-point boundary value problem of the form

y00 = f (t,y,y0), y(a) = α, y(b) = β. (2)

If f is locally Lipschitz in the last two variables then by the Picard-Lindelof¨ Theorem, for any γ ∈ R the initial value problem

y00 = f (t,y,y0), y(a) = α, y0(a) = γ (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) = α, u(a) = γ 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) = α + γ(t − a) + (t − s)f (s,y(s),y0(s))ds, (5) a

which we then solve, when evaluated at t = b, for γ:

Z b γ = 1 β − α − (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 γ. Thus the iterates are

y[0](t) ≡ α

[0] β−α u (t) ≡ b−a (7a) [0] β−α γ ≡ b−a

and Z t y[k+1](t) = α + u[k](s)ds a Z t u[k+1](t) = γ[k] + f (s,y[k](s),u[k](s))ds (7b) a 1 Z b γ[k+1] = β − α − (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 α, β ∈ R the boundary value problem

y00 = f (t,y,y0), y(a) = α, y(b) = β (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 γ = 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 γ[0] = 1.2 and   y[k+1](t)  0   [k+1]  = 1 R 1.2 [k] (11) u (t)  1.2 (β − α + 0 (1.2 − s)v (s)ds v[k+1](t) 1

 [k]  Z t u (s) +  −v[k](s)  ds, 0   −2u[k](s)v[k](s)

where we have shifted the update of γ and incorporated it into the update of u[k](t). The ﬁrst eight iterates of γ are:

γ[1] = −0.24594, γ[2] = 0.16011, γ[3] = 0.19297, γ[4] = 0.04165,

γ[5] = −0.04272, γ[6] = −0.04012, γ[7] = −0.00923, γ[8] = 0.01030,

[8] The maximum error in the approximation is |yexact − y |sup ≈ 0.0065. 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

EXAMPLE

In this example the right hand side is not autonomous and does not satisfy a global Lipschitz condition. Consider the boundary value problem

1 43 y00 = 32 + 2t3 − yy0 , y(1) = 17, y(3) = (12) 8 3

3 16 The unique solution is y(t) = t + t . We have y[0](t) ≡ 17 4 u[0](t) ≡ − 3 (13a) 4 γ[0] = − 3 and ! ! y[k+1](t) 17 Z t u[k](s) [k+ ] = [k] + 1 3 [k] [k] ds, u 1 (t) γ 1 4 + s − y (s)u (s) 4 (13b) 4 1 Z 3 1 γ[k+1] = − − (3 − s)(4 + s3 − y[k](s)u[k](s))ds. 3 2 1 4

[12] 76 After n = 12 iterations, γ = −8.443, whereas the exact value is γ = − 9 ≈ −8.444. As illustrated by the error plot, see the [12] Figure in the next slide for the maximum error in the twelfth approximating function is |yexact − y |sup ≈ 0.0012. 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

MIXED TYPE BOUNDARY CONDITIONS

The ideas developed at the beginning can also be applied to two-point boundary value problems of the form

y00 = f (t,y,y0), y0(a) = γ, y(b) = β, (14)

so that in equation (3) the constant γ is now known and α = y(a) is unknown. Thus in (5) we evaluate at t = b but now solve for α instead of γ, obtaining in place of (6) the expression

Z b α = β − γ(b − a) − (b − s)f (s,y(s),y0(s))ds. a

In the Picard iteration scheme we now successively update an approximation of α starting with some initial value α0. The iterates are [0] y (t) ≡ α0 u[0](t) ≡ γ (15a) [0] α = α0

and Z t y[k+1](t) = α[k] + u[k](s)ds a Z t u[k+1](t) = γ + f (s,y[k](s),u[k](s))ds (15b) a Z b α[k+1] = β − γ(b − a) − (b − s)f (s,y[k](s),u[k](s))ds. 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

Consider the boundary value problem

y00 = −y0e−y, y0(0) = 1, y(1) = ln2 (16)

The exact solution is y(t) = ln(1 + t), for which y(0) = ln1 = 0. Introducing the dependent variable u = y0 as always to obtain the equivalent ﬁrst order system y0 = u, u0 = −ue−y and the variable v = e−y to replace the transcendental function with a polynomial we obtain the expanded system

y0 = u u0 = −uv v0 = −uv

with initial conditions y(0) = α, u(0) = γ, v(0) = e−α,

so that, making the initial choice α0 = 1 [0] y (t) ≡ α0 u[0](t) ≡ 1 (17)

v[0](t) ≡ e−α0

and 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[k+1](t)  R 1 [k] [k]  ln2 − 1 + 0 (1 − s)u (s)v (s)ds  [k+1]  u (t) =  1  (18) v[k+1](t) e−α0

 [k]  Z t u (s) + −u[k](s)v[k](s) ds, 0   −u[k](s)v[k](s)

where we have shifted the update of α and incorporated it into the update of y[k](t). We list the ﬁrst eight values of α[k] to show the rate of convergence to the exact value α = 0.

α[1] = −0.00774, α[2] = 0.11814, α[3] = 0.07415, α[4] = 0.08549,

α[5] = 0.08288, α[6] = 0.08339, α[7] = 0.08330, α[8] = 0.08331 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

MIXED TYPE BOUNDARY CONDITIONS:ASECOND APPROACH

In actual applications the Algorithm converges relatively slowly because of the update at every step of the estimate of the initial value y(a) rather than of the initial slope y0(a). A different approach that works better in practice with a problem of the type

y00 = f (t,y,y0), y0(a) = γ, y(b) = β, (19)

is to use the relation Z b y0(b) = y0(a) + f (s,y(s),y0(s))ds (20) a

between the derivatives at the endpoints to work from the right endpoint of the interval [a,b], at which the value of the solution y is known but the derivative y0 unknown. That is, assuming that (19) has a unique solution and letting the value of its derivative at t = b be denoted δ, it is also the unique solution of the initial value problem

y00 = f (t,y,y0), y(b) = β, y0(b) = δ. (21) 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

We introduce the new dependent variable u = y0 to obtain the equivalent system

y0 = u u0 = f (t,y,u)

with initial conditions y(b) = β and y0(b) = δ and apply Picard iteration based at t = b, using (20) with y0(a) = γ to update the 0 approximation of y (b) are each step. Choosing a convenient initial estimate δ0 of δ, the successive approximations of the solution of (21), hence of (19), are given by y[0](t) ≡ β [0] u (t) ≡ δ0 (22a) [0] δ = δ0

and Z t y[k+1](t) = β + u[k](s)ds b Z t u[k+1](t) = δ[k] + f (s,y[k](s),u[k](s))ds (22b) b Z b δ[k+1] = γ + f (s,y[k](s),u[k](s))ds. 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

EXAMPLE

Reconsider the boundary value problem

y00 = −y0e−y, y0(0) = 1, y(1) = ln2 (23)

with exact solution y(t) = ln(1 + t). The relation (20) is

Z 1 y0(1) = 1 − y0(s)e−y(s) ds. (24) 0

We set u = y0 and v = e−y to obtain the expanded system

y0 = u u0 = −uv v0 = −uv

Since, our standard way is to have initial condition at t = 0. Therefore we use change of variable τ = 1−t and Example becomes

y00 = y0e−y, y(0) = ln2, y0(1) = −1 (25)

with exact solution y(t) = ln(τ − 2). The relation (20) is

Z 1 y0(1) = −1 + y0(s)e−y(s) ds. (26) 0 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

We set u = y0 and v = e−y to obtain the expanded system

y0 = u u0 = uv v0 = −uv

y[0](t) ≡ ln2

u[0](t) ≡ −1 (27a) [0] 1 v (t) ≡ 2 δ[0] ≡ −11

and  [k+1]     [k]  y (t) ln2 Z t u (s)  [k+1]  [k]  [k] [k]  u (t) = δ  +  u (s)v (s)  ds, [k+1] 1 0 [k] [k] v (t) 2 −u (s)v (s) (27b) Z 1 δ[k+1] = −1 + u[k](s)v[k](s)ds. 0

0 1 [k] The exact value of δ = y (1) is 2 . The ﬁrst eight values of δ are:

δ[0] = −0.50000, δ[1] = 0.41667, δ[2] = 0.50074, δ[3] = 0.51592,

δ[4] = 0.49987, δ[5] = 0.49690, δ[6] = 0.50003, δ[7] = 0.50060, δ[8] = 0.50000 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

LONG INTERVALS

Consider the following Boundary Value Ordinary Differential Equation,

w00(t) = f (t,w(t),w0(t)), a ≤ t ≤ b (28) w(a) = α, w(b) = β

Here 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. For simplicity we divided interval [a,b] into three subintervals [a,t1], [t1,t2] and [t2,b]. Where t0 = a < t1 < t2 < t3 = b. Where h = ti − ti−1 for 1 ≤ i ≤ 3 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

We would like to show that the boundary value problem (28) on the long interval has a unique solution if the following Initial value problems can simultaneously generate recursively a convergence sequence of y(t) restricted to each subintervals.

00 0 w1 (t) = f (t,w1(t),w1(t)), a ≤ t ≤ t1 0 (29) w1(a) = α, w1(a) = γ1

00 0 w2 (t) = f (t,w2(t),w2(t)), t1 ≤ t ≤ t2 0 (30) w2(t1) = β1, w2(t1) = γ2 and

00 0 w3 (t) = f (t,w3(t),w3(t)), t2 ≤ t ≤ b 0 (31) w3(t2) = β2, w3(t2) = γ3

We convert interval [ti−1,ti] to [0,h] by τ = t − ti−1. This transformation can help us to simplify the proof. Hence (29) becomes 00 0 y1 (τ) = f1(τ,y1(τ),y1(τ)), 0 ≤ τ ≤ h 0 (32) y1(0) = α, y1(0) = γ1 (30) becomes 00 0 y2 (τ) = f2(τ,y2(τ),y2(τ)), 0 ≤ τ ≤ h 0 (33) y2(0) = β1, y2(0) = γ2 and (31) becomes

00 0 y3 (τ) = f3(τ,y3(τ),y3(τ)), 0 ≤ τ ≤ h 0 (34) y3(0) = β2, y3(0) = γ3 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 initial conditions for each of the equations (32), (33) and (34) will be obtained so that the following conditions are satisﬁed

y1(h;β1,γ1) = y2(0;β1;γ2) Continuity Condition y2(h;β2,γ2) = y3(0;β2;γ3) Continuity Condition 0 0 y1(h;β1,γ1) = y2(0;β1;γ2) Smoothness Condition (35) 0 0 y2(h;β2,γ2) = y3(0;β2;γ3) Smoothness Condition y3(h;β2;γ3) = β

These conditions will be satisﬁed automatically inside of the algorithm. The recursion initialization becomes [0] y1 (τ) ≡ α [0] β − α u (τ) ≡ 1 b − a [0] [0] γ ≡ u 1 1 (36) [0] [0] β1 ≡ u1 h + α [0] [0] u2 (τ) ≡ γ1 [0] [0] u3 (τ) ≡ γ1 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

! ! ! [k+1] α Z τ [k] y1 (τ) u1 (s) [k+ ] = [k] + [k+ ] [k] ds (37a) 1 γ 0 1 u1 (τ) 1 f1(s,y1 (s),u1 (s))

[k] [k+1] [k] [k+1] φ1 ≡ y1 (h), γ2 ≡ u1 (h) (37b)

[k+1] ! [k]! Z τ [k] ! y2 (τ) φ1 u2 (s) [k+ ] = [k] + [k+ ] [k] ds (37c) 1 0 1 u2 (τ) γ2 f2(s,y2 (s),u2 (s))

[k] [k+1] [k] [k+1] φ2 ≡ y2 (h), γ3 ≡ u2 (h) (37d)

[k+1] ! [k]! Z τ [k] ! y3 (τ) φ2 u3 (s) [k+ ] = [k] + [k+ ] [k] ds (37e) 1 0 1 u3 (τ) γ3 f3(s,y3 (s),u3 (s))

[k] [k] Z h [k+1] [k+1] β = β − γ h − (h − s)f3(s,y (s),u (s))ds 2 3 0 3 3 [k+1] [k] [k] Z h [k+1] [k+1] β = β − γ h − (h − s)f2(s,y (s),u (s))ds (37f) 1 2 2 0 2 2 Z h [k+1] 1 [k+1] [k+1] γ = h β1 − α − (h − s)f1(s,y (s),u (s))ds . 1 0 1 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

Example Consider the boundary value problem, we

1 y00 = 2y3, y(0) = − , y(2) = β (38) 4

Our goal is to divide a long intervals into n subintervals, with relatively small length, that satisﬁes our original Theorem. We 5 5 7 7 divided the interval into three subintervals [0, 4 ] ∪ [ 4 , 4 ] ∪ [ 4 ,2]. For n = 18. The exact value of γ = −0.0625. The ﬁrst 10 values of γ[k] are:

γ[1] = −0.12500, γ[2] = −0.10156, γ[3] = −0.06154, γ[4] = −0.04716, γ[5] = −0.06917,

γ[6] = −0.06903, γ[7] = −0.05852, γ[8] = −0.05948, γ[9] = −0.06515, γ[10] = −0.06366, 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

VOLTERRA INTEGRAL

A Volterra equation of the second kind is an equation of the form

Z t y(t) = ϕ(t) + K(t,s)f (s,y(s))ds (39) 0 where ϕ, K, and f are known functions of suitable regularity and y is an unknown function. Such equations lend themselves to solution by successive approximation using Picard iteration, although in the case that the known functions are not polynomials the process can break down when quadratures that cannot be performed in closed form arise. If the “kernel“ K is independent of the ﬁrst variable t then the integral equation is equivalent to an initial value problem. Indeed, in precisely the reverse of the process. 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

We have that Z t y(t) = ϕ(t) + k(s)f (s,y(s))ds 0 is equivalent to

y0(t) = ϕ0(t) + k(t)f (t,y(t)), y(0) = ϕ(0).

In this chapter we provide a method for introducing auxiliary variables in (39), in the case that K factors as K(t,s) = j(t)k(s), in such a way that it embeds in a vector-valued polynomial Volterra equation, thus extending the Parker-Sochacki method to this setting and thereby obtaining a computationally efﬁcient method for closely approximating solutions of (39). 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

EXAMPLE (VIE)

Linear

Consider the following second kind linear volterra integral.

Z t 2 + cos(t) y(t) = exp(t)sin(t) + y(s)ds (40) 0 2 + cos(s)

 φ(t) = exp(t)sin(t)     k(t,s) = 2+cos(t) (41)  2+cos(s)    f (t,y(t)) = y(t)

The exact solution is y(t) = exp(t)sin(t) + exp(t) 2 + cos(t) ln(3) − ln2 + cos(t) (42) 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

EXAMPLE

( y(t) = exp(t)sin(t) + (2 + cos(t))R t 1 y(s)ds 0 2+cos(s) (43) y(0) = exp(0)sin(0) = 0

We deﬁne the following variables

v1 = exp(t), v1(0) = 1

v2 = cos(t), v2(0) = 1

v3 = sin(t), v3(0) = 0 (44)

v4 = 2 + v2, v4(0) = 3

1 1 v5 = , v5(0) = v4 3

0 I v1 = v1 0 I v2 = −v3 0 I v3 = v2 0 0 I v4 = v2 = −v3 −v0 0 4 2 I v = = v3v 5 v2 5 4 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

EXAMPLE

R t I v1(t) = v1(0) + 0 v1(s)ds R t I v2(t) = v2(0) − 0 v3(s)ds R t I v3(t) = v3(0) + 0 v2(s)ds R t I v4(t) = v4(0) − 0 v3(s)ds R t 2 I v5(t) = v5(0) + 0 v3(s)v5ds

I y[1] = 0 = α [1] I v1 = exp(0) = 1 [1] I v2 = cos(0) = 1 [1] I v3 = sin(0) = 0 [1] [1] I v4 = 2 + v2 = 3 [1] 1 I v5 = 3 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

EXAMPLE VIE Thus the iterates are

[k+1] [k] [k] [k] R t [k] [k] I y = q1 v3 + v4 0 v5 y ds [k+1] R t [k] I v1 = 1 + 0 v1 ds [k+1] R t [k] I v2 = 1 − 0 v3 ds [k+1] R t [k] I v3 = 0 v2 ds [k+1] R t [k] I v4 = 3 − 0 v3 ds [k+1] 1 R t [k] [k] 2 I v5 = 3 + 0 v3 (v5 ) ds

The approximation of solution for the ﬁrst 4 iterations

y[2] = 0

y[3] = t2 + t 1 1 y[4] = − t7 − t6 − 1/45t5 − 1/24t4 + 5/6t3 + 3/2t2 + t 180 144 1 1 59 41 y[5] = t16 + t15 + t14 + t13 (45) 9797760 7278336 29393280 13343616 1 1 1 1 1 1 − t12 − t11 − t10 − t9 − t8 − t7 87480 42768 2880 1512 480 336 49 − t6 − 1/8t5 + 1/6t4 + 5/6t3 + 3/2t2 + t 1080 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

EXAMPLE VIE

For n = 5

[n+1] yapprox = y , Exact = y(t) 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

EXAMPLE VIE

By increasing n tto n = 7

EXAMPLE VIE Non-Linear

Biazar on his paper [11] has introduced the following second kind non-linear volterra integral.

 φ(t) = 1/2sin(2t)     k(t,s) = cos(s − t) (46)    f (t,y) = y2 

THEEXACTSOLUTIONIS y(t) = sin(t) (47)

To solve this non-linear Volterra integral we set up the integral

 y(t) = 1 sin(2t) + 3 R t cos(s − t)y2(s)ds  2 2 0   (48) y(0) = 1 sin(2(0)) = 0  2   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

EXAMPLE VIE

For n = 5

Error = |y[n+1] − y(t)| 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

EXAMPLE VIE

By increasing n to n = 7 we have

THANKYOU

Thank you 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

(http://en.wikipedia.org/wiki/Parker-Sochacki-method) Herbert B. Keller; Numerical Methods For Two Point Boundary Value Problems David C. Carothers , G. Edgar Parker, James S. Sochacki, Paul G. Warne; Some Properties Of Solutions To Polynomial Systems Of Differential Equations Abdul Majid Wazwaz;First Course in Integral Equations, (December 1997) Publisher: World Scientiﬁc Publishing Company, New Jersey and Singapore http : //en.wikipedia.org/wiki http : //en.wikipedia.org/wiki/Paul du Bois − Reymond http : //calculus7.org/2012/06/18/ivp − vs − bvp/ G. Edgar Parker And James S. Sochacki; A Picard-Maclaurin Theorem For Initial Value Pdes 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

David C. Carothers , G. Edgar Parker, James S. Sochacki, D. A. Warne, P. G. Warne; Explicit A-Priori Error Bounds and Adaptive Error Control for Approximation of Nonlinear Initial Value Differential Systems David Carothers, William Ingham, James Liu, Carter Lyons, John Maraﬁno, G. Edgar Parker, David Pruett, Joseph Rudmin, James S.Sochacki, Debra Warne, Paul Warne, ···; An Overview Of The Modiﬁed Picard Method G. Edgar Parker, James S.Sochacki; Implementing Picard Methods Jafar Biazar and Mostafa Eslami; Homotopy Perturbation and Taylor Series for Volterra Integral of Second Kind. Sohrab Effati, Mohammad Hadi Noori Skandari; Optimal Control Approach for Solving Linear Volterra Integral Equations Dennis Aebersold ; Integral Equations in Quantum Chemistry 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

Peter Linz; Numerical Methods For Volterra Integral Equations Of The First Kind Alexander John Martyn Little; Analytical And Numerical Solution Methods For Integral Equations Using Maple Symbolic Algebriac Package Sivaji Ganesh Sista; Ordinary Differential Equations, Lecture Notes Chapter 3 Denise Gutermuth; Picard’s Existence and Uniqueness Theorem, Lecture Notes