A numerical technique based on reproducing kernel methods for the exact
solution of linear Volterra integral equations system of the second kind is
given. The traditional reproducing kernel method requests that operator a satisfied linear operator equation
The purpose of this paper is to solve a system of linear Volterra integral equations
In (
Since the reproducing kernel space
The reproducing kernel space
The inner product and the norm are equipped with
The reproducing kernel space
The inner product and the norm are equipped with
The proof of Theorems
Hilbert space
It is easy to prove that
Consider the
Define operator
Let
Take
From this fact, it holds that
for every
Since (
We arrange
The exact solution of (
Assume that
The approximate solution of (
Obviously,
Note that
If
Taking nodes
Consider the following system of Volterra integral equations of the second kind [
Absolute errors for Example
Nodes |
Errors |
Errors |
Errors |
Errors |
---|---|---|---|---|
0.0 | 0 |
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0 |
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0.1 |
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0.2 |
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0.3 |
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0.4 |
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0.5 |
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0.6 |
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0.7 |
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0.8 |
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0.9 |
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1.0 |
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Consider the following system of linear Volterra integral equations of the second kind [
Absolute errors for Example
Nodes |
Errors |
Errors |
Errors |
Errors |
---|---|---|---|---|
0.0 | 0 |
|
0 |
|
0.1 |
|
|
|
|
0.2 |
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0.3 |
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0.4 |
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0.5 |
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0.6 |
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0.7 |
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0.8 |
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0.9 |
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1.0 |
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In this paper, we modify the traditional reproducing kernel method to enlarge its application range. The new method named MRKM is applied successfully to solve a system of linear Volterra integral equations. The numerical results show that our method is effective. It is worth to be pointed out that the MRKM is still suitable for solving other systems of linear equations.
The research was supported by the Fundamental Research Funds for the Central Universities.