Based on recent progress on moment problems, semidefinite optimization approach is proposed for estimating upper and lower bounds on linear functionals defined on solutions of linear integral equations with smooth kernels. The approach is also suitable for linear integrodifferential equations with smooth kernels. Firstly, the primal problem with smooth kernel is converted to a series of approximative problems with Taylor polynomials obtained by expanding the smooth kernel. Secondly, two semidefinite programs (SDPs) are constructed for every approximative problem. Thirdly, upper and lower bounds on related functionals are gotten by applying SeDuMi 1.1R3 to solve the two SDPs. Finally, upper and lower bounds series obtained by solving two SDPs, respectively infinitely approach the exact value of discussed functional as approximative order of the smooth kernel increases. Numerical results show that the proposed approach is effective for the discussed problems.
Semidefinite optimization has been successfully applied to deal with many important problems [
In this paper, we will extend the polynomial kernels in [
The rest of this paper is organized as follows. In Section
In this section, we propose semidefinite optimization method for estimating guaranteed bounds on linear functionals defined on solutions of Volterra integral equation of the second kind with smooth kernel.
Throughout the work, we suppose that related integral equations make unique solutions exist in the distribution space
Equation (
We expand the kernel
Therefore, (
For simplicity, we rewrite
Further,
In [
Suppose that the solution
We define
Multiplying (
By (
Substituting (
Because the solution
It is obvious that
Denote
By the method in [
We can obtain the following positive semidefinite matrices:
Assuming that the testing function with the highest degree is
Denote by
When
When
Obviously,
The proposed method is also suitable for other linear integral and integrodifferential equations with smooth kernels.
In general,
The semidefinite constraints (
In practical applications, for reducing computation amounts of Algorithm
In some cases,
In this section, we give four examples to illustrate the effectiveness of Algorithm
Computing
The exact solution of (
Multiplying (
In this example,
Substituting (
We construct the following two SDPs:
Letting
Upper and lower bounds for Example
UNOb  LLFoS  ULFoS  ELFoS  Error 


0.842  0.855  0.841  0.014 

0.382  0.396  0.382  0.014 

0.240  0.253  0.239  0.014 

0.172  0.186  0.172  0.014 

0.134  0.147  0.133  0.014 

0.109  0.122  0.108  0.014 

0.092  0.105  0.091  0.014 

0.079  0.092  0.078  0.014 

0.069  0.083  0.069  0.014 

0.062  0.075  0.061  0.014 

0.056  0.069  0.055  0.014 






0.028  0.042  0.029  0.013 
In Table
Upper and lower bounds for Example
UNOb  LLFoS  ULFoS  ELFoS  Error 


0.84148  0.84148  0.84147  0.00001 

0.38180  0.38180  0.38177  0.00003 

0.23916  0.23916  0.23913  0.00003 

0.17176  0.17176  0.17174  0.00002 

0.13309  0.13309  0.13308  0.00001 

0.10823  0.10823  0.10822  0.00001 

0.09100  0.09100  0.09098  0.00002 

0.07839  0.07839  0.07837  0.00002 

0.06878  0.06878  0.06877  0.00001 

0.06123  0.06123  0.06122  0.00001 

0.05515  0.05515  0.05514  0.00001 






0.02750  0.02750  0.02750  0.00000 






0.01826  0.01826  0.01826  0.00000 






0.01366  0.01366  0.01366  0.00000 
Upper and lower bounds for Example
UNOb  LLFoS  ULFoS  ELFoS  Error 


1.333335  1.333335  1.333333  0.000000 

0.916669  0.916669  0.916667  0.000002 

0.700002  0.700002  0.700000  0.000002 

0.566668  0.566668  0.566667  0.000001 

0.476192  0.476192  0.476190  0.000002 

0.410716  0.410716  0.410714  0.000002 

0.361113  0.361113  0.361111  0.000002 

0.322224  0.322224  0.322222  0.000002 

0.290910  0.290910  0.290909  0.000001 

0.265153  0.265153  0.265152  0.000000 

0.243591  0.243591  0.243590  0.000000 






0.134388  0.134388  0.134387  0.000000 






0.092804  0.092804  0.092803  0.000000 






0.070875  0.070875  0.070875  0.000000 
Upper and lower bounds for Example
UNOb  LLFoS  ULFoS  ELFoS  Error 


0.666762  0.666762  0.666667  0.000095 

0.291711  0.291711  0.291667  0.000044 

0.183362  0.183362  0.183333  0.000029 

0.133354  0.133354  0.133333  0.000021 

0.104778  0.104778  0.104762  0.000016 

0.086323  0.086323  0.086310  0.000013 

0.073424  0.073424  0.073413  0.000009 

0.0638985  0.063985  0.063889  0.000014 

0.0565740  0.0566574  0.056566  0.000008 

0.050765  0.050765  0.050758  0.000007 

0.046044  0.046044  0.046037  0.000003 






0.023907  0.023907  0.023904  0.000003 






0.016162  0.016162  0.016159  0.000003 






0.012210  0.012210  0.012209  0.000001 
Upper and lower bounds for Example
UNOb  LLFoS  ULFoS  ELFoS  Error 


1.7185  1.7185  1.7183  0.0002 

1.0002  1.0002  1.0000  0.0002 

0.7185  0.7185  0.7183  0.0002 

0.5636  0.5636  0.5634  0.0001 

0.4647  0.4647  0.4645  0.0001 

0.3957  0.3957  0.3956  0.0001 

0.3448  0.3448  0.3447  0.0001 

0.3056  0.3056  0.3055  0.0001 

0.2744  0.2744  0.2744  0.0000 

0.2491  0.2491  0.2490  0.0001 

0.2281  0.2280  0.2280  0.0000 






0.1238  0.1238  0.1238  0.0000 






0.0850  0.0850  0.0850  0.0000 






0.0648  0.0648  0.0648  0.0000 
In order to increase the precision of numerical results of Example
Differentiating both sides of (
Use the two polynomials
Substituting (
Let
Substituting (
According to
We construct the SDPs:
Letting
From Table
Computing
The exact solution of the equation is
Multiplying (
Because
We construct the following SDPs:
Letting
Numerical results in Table
Computing
The exact solution of (
Chebyshev polynomial with degree 3 of
Define
For
We construct the following SDPs:
Letting
Obviously, the numerical results in Table
Computing
The exact solution of (
Integrating (
Replace
Numerical results in Table
In this paper, we have presented the semidefinite optimization method for providing guaranteed bounds on linear functionals defined on solutions of linear integral equations with smooth kernels. Four examples show that the proposed approach is effective for estimating bounds on linear integral and integrodifferential equations with smooth kernels. The proposed approach requires that the related integral equation is linear. It cannot be directly applied to solve the nonlinear integral equation. So next work is to improve the proposed method, so that it can handle nonlinear problems.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the anonymous reviewers for their helpful comments. This work is partially supported by the National Natural Science Foundation, Grants nos. 61202397 and 11301445, the Research Foundation of Education Bureau of Hunan Province, Grant no. 13B121, the Research Foundation for Doctoral Program of Higher Education of China Grant no. 20114301120001, and the Science and Technology Project of Hunan Province.