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Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of integration including fractional integral equations. Numerical tests verify the validity of the proposed schemes.

The Volterra integral equations arise in many scientific and engineering fields such as the population dynamics, spread of epidemics, semi-conductor devices, vehicular traffic, the theory of optimal control, the kinetic theory of gases and economics [

In this work, we consider the general nonlinear Volterra integral equation of the second kind

where it permits weak singularity at the limits of integration.

The specific conditions under which a solution exists for the nonlinear Volterra integral equation are considered in [

The Picard iteration method, or the successive approximations method, is a direct and convenient technique for the resolution of differential equations. This method solves any problem by finding successive approximations to the solution by starting with the zeroth approximation. The symbolic computation applied to the Picard iteration is considered in [

In this work, we concern on the numerical Picard iteration methods for nonlinear Volterra integral Equation (1). By using the proposed methods, we treat the involved integrals numerically and enlarge the effective region of convergence of the Picard iteration. The rest of the paper is organized as follows. In Section 2, the scheme in a single interval is considered, and the validity of the method is verified by some numerical tests. Basing on the scheme proposed in Section 2, we devise a multistage algorithm in Section 3 for enlarging the convergence region. In Section 4, an algorithm is introduced for problems with some singularity. To show the effectiveness of the proposed algorithms, we perform some numerical results.

The Picard iteration scheme for the considered Equation (1) reads [

The Picard iteration scheme has been applied in almost each textbook on differential equations to mainly prove the existence and uniqueness of solutions. It is direct and easily learned for numerical calculation.

Assume the recursion scheme is convergent for

At

Treating the integral involved in (4) by numerical quadrature formulas, we have the numerical Picard iteration scheme for (1) over

where

Numerical results are given to validate the proposed scheme. Let us start with an example in which the inte- grand

Example 1 Consider the initial value problem (IVP) for the nonlinear differential equation

This IVP has the exact solution

The equivalent integral equation of the IVP is

Denote

The relative errors

numerical integration are needed. For each fixed N, iterations stop when

Next we give an example with t-dependent integrand.

Example 2 Consider the pendulum equation

The exact solution can be expressed in terms of the Jacobi elliptic function

Integrating the differential equation in (7) yields

Take

the results of the first 5 iterations and the errors at T for each iteration. It confirms the validity of the scheme (5), (6) for equations with general integrand f.

What’s different from Example 1 is that, at T, the results of the second and the third iterations are even worse than the first one. However, it can be noticed that, in the interval closer to t = 0, for example

It’s well-known that the convergence of the Picard iteration is constrained in some interval. Then how can we get the numerical solution to the integral Equation (1) when t is outside the interval of convergence? We will take advantage of the multistage method and design a scheme by which the considered problem can be solved interval by interval. For example, the Equation (1) is considered on_{1} < T. For achieving the numerical result at T, we can regard the problem on

Denote the time interval considered for (1) by

where

Suppose the equation has been solved on

Now we consider the solution on

we have for

the right hand side of which will be analyzed below.

・ An approximation

・ The second part, with the approximations of

where the corresponding weights for numerical integration on

・

Denoting

(9) leads to a new equation, which is similar to the considered problem (1),

namely,

Using (5), (6) over

We conclude the previous analysis as an algorithm.

Algorithm 1 Choose the algorithm’s parameters: number of subintervals

Step 1. For

- the uniformly distributed nodes and corresponding weights

- the weights

Step 2. For k = 1, solve (12). Note that the first term of

Step 3. Recursively solve (12) for

- Calculate

- The initial value of iteration:

- For

Here, we perform a numerical test to examine the effectiveness of Algorithm 1 and compare it with the scheme in single interval (2).

Example 3 Consider the Lane-Emden equation

The exact solution is

The equivalent integral form of the Lane?Emden equation is [

First, taking T = 4,

Consider the underlying problem for larger T by Algorithm 1. The time interval

where

N | K = 3 | K = 4 | K = 6 | K = 12 | ||||
---|---|---|---|---|---|---|---|---|

Error | Order | Error | Order | Error | Order | Error | Order | |

10 | 1.814e−3 | 9.131e−4 | 3.771e−4 | 9.040e−5 | ||||

20 | 3.771e−4 | −2.27 | 2.070e−4 | −2.14 | 9.040e−5 | −2.06 | 2.237e−5 | −2.01 |

30 | 1.625e−4 | −2.08 | 9.040e−5 | −2.04 | 3.987e−5 | −2.02 | 9.922e−6 | −2.01 |

40 | 9.040e−5 | −2.04 | 5.054e−5 | −2.02 | 2.237e−5 | −2.01 | 5.578e−6 | −2.00 |

In fact, from the errors reported in the table, the convergence order

It’s an interesting phenomenon observed from

In recent years, the fractional differential or integral equations are much involved. In fact, fractional integral is a class of integration with weak singular kernel. So many fractional differential and integral equations can be equivalently expressed by the singular Volterra integral equation of the second kind. Let us consider such an integral equation with some singularity.

Example 4 Consider the singular Volterra integral equation [

The exact solution is

A simple idea is to avoid the value of the integrand at s = t in the numerical integration, so an alternative is to

integrate with compound rectangular formula. The only things we need to do are changing the nodes of numerical integration and generating approximations for the values of

nodes

For

Denote

in which

Thus, (12) becomes

We present the following algorithm.

Algorithm 2 Choose the algorithm’s parameters: number of subintervals

Step 1. For

- the nodes

- the integral nodes and weights

- the weights

Step 2. Solve (16) for k = 1. As in Algorithm 1, since

- For

- For

where

Step 3. Recursively solve (16) for

- For

- For

where

Now, we come back to Example 4. Taking

Remark 1. Algorithm 2 is devised not especially for singular problems. It’s also valid for regular problems. For instance, we recalculate Example 1 with K = 2 and N = 5, 10, 20, 40, 80, 160. Errors and convergence rates respect to N are reported in

In this work, Picard iteration methods with numerical integration are devised for the second kind nonlinear Volterra integral equations. The Picard iteration method solves the considered nonlinear equation explicitly, while the multistage scheme solves it interval by interval and enlarges the convergence region of the Picard iteration method. Numerical results validate the proposed schemes and algorithms and reveal that the schemes are of order

N | 5 | 10 | 20 | 40 | 80 | 160 |
---|---|---|---|---|---|---|

Error | 5.807e−0 | 1.376e−0 | 3.394e−1 | 8.459e−2 | 2.113e−2 | 5.281e−3 |

Order | −2.08 | −2.02 | −2.00 | −2.00 | −2.00 |

What should be noticed is that the errors reported in the numerical results decrease exponentially respect to times of iteration n (for example, through simple calculation, we can observe from

This work was supported by the Natural Science Foundation of Shanghai (No. 14ZR1440800) and the Innovation Program of the Shanghai Municipal Education Commission (No. 14ZZ161).

Lian Chen,Junsheng Duan, (2015) Multistage Numerical Picard Iteration Methods for Nonlinear Volterra Integral Equations of the Second Kind. Advances in Pure Mathematics,05,672-682. doi: 10.4236/apm.2015.511061