Numerical modelling of qualitative behaviour of solutions to convolution integral equations. (English) Zbl 1125.65118
The authors focus on the qualitative behaviour of solutions to linear integral equations of the form
\[ y(t)=g(t)+ \int_0^t k(t-s)y(s)\,ds. \]
The kernel \(k\) is assumed to be either integrable or of exponential type. The particular emphasis of the paper is on preserving qualitative properties of exact solutions under (discrete) numerical approximations. Analytical results for the exact solutions of some convolution integral equations are obtained and for their numerical approximations. Section 2 includes definitions and theory relevant to the research presented, including the definition of a function which is positive definite of exponential type.
Section 3 focuses on the resolvent kernel. A proposition relating to the long-term behaviour of solutions to the equation \(y(t)=g(t)+k*y(t)\) is presented and proved. In section 4 the equation \(y(t)=g(t)+\int_0^t \frac{s^{\mu-1}}{t^\mu}y(s)\,ds\) (which has a non-unique solution for \(0< \mu <1\)) is expressed in the form of a convolution equation with kernel \(e^{-\mu (\tau -\zeta)}\), thus enabling analysis of the asymptotic behaviour of \(y\) as \(t \rightarrow \infty\).
Sections 5 and 6 concern numerical methods and results. Relevant literature is cited. The authors state and prove a theorem relating to sequences of exponential type and give results that show that the exponential type of a solution may be preserved in the numerical scheme for all \(h>0\). The paper concludes with an illustrative example to demonstrate that the numerical results reflect the analytical properties of the solution.
\[ y(t)=g(t)+ \int_0^t k(t-s)y(s)\,ds. \]
The kernel \(k\) is assumed to be either integrable or of exponential type. The particular emphasis of the paper is on preserving qualitative properties of exact solutions under (discrete) numerical approximations. Analytical results for the exact solutions of some convolution integral equations are obtained and for their numerical approximations. Section 2 includes definitions and theory relevant to the research presented, including the definition of a function which is positive definite of exponential type.
Section 3 focuses on the resolvent kernel. A proposition relating to the long-term behaviour of solutions to the equation \(y(t)=g(t)+k*y(t)\) is presented and proved. In section 4 the equation \(y(t)=g(t)+\int_0^t \frac{s^{\mu-1}}{t^\mu}y(s)\,ds\) (which has a non-unique solution for \(0< \mu <1\)) is expressed in the form of a convolution equation with kernel \(e^{-\mu (\tau -\zeta)}\), thus enabling analysis of the asymptotic behaviour of \(y\) as \(t \rightarrow \infty\).
Sections 5 and 6 concern numerical methods and results. Relevant literature is cited. The authors state and prove a theorem relating to sequences of exponential type and give results that show that the exponential type of a solution may be preserved in the numerical scheme for all \(h>0\). The paper concludes with an illustrative example to demonstrate that the numerical results reflect the analytical properties of the solution.
Reviewer: Pat Lumb (Chester)
MSC:
| 65R20 | Numerical methods for integral equations |
| 45D05 | Volterra integral equations |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 45M05 | Asymptotics of solutions to integral equations |
Keywords:
Volterra integral equations; resolvent kernel; convolution integral equation; asymptotic behaviour; numerical resultsReferences:
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