45D05 Volterra integral equations

Overview

45D05 treats volterra integral equations within Volterra integral equations. Typical questions involve existence, uniqueness, representation formulas, inversion, and stability of solution operators in analytic and applied settings.

Related Wikipedia Page

Wikipedia search: Volterra integral equations

Useful Links

• Wikipedia search: Volterra integral equations
• Encyclopaedia of Mathematics
• arXiv: Classical Analysis and ODEs (math.CA)
• Journal of Integral Equations and Applications
• Nonlinear Analysis
• Applied Mathematics and Computation

Key Ideas

• Canonical formulations and operator-theoretic viewpoints for volterra integral equations
• Kernel properties, transform methods, and regularity assumptions
• Analytic and numerical criteria for solvability and stability

Typical Uses

Used to reformulate differential and boundary-value models as operator equations, derive qualitative properties of solutions, and support approximation schemes.

Applications

• Potential theory, wave/heat propagation, and inverse-problem modeling
• Numerical simulation pipelines where integral formulations improve conditioning
• Systems and control contexts using convolution and memory-type operators

References

• Gripenberg, G., Londen, S.-O. and Staffans, O. (1990) Volterra Integral and Functional Equations. Cambridge: Cambridge University Press.
• Brunner, H. (2017) Volterra Integral Equations: An Introduction to Theory and Applications. 2nd edn. Cambridge: Cambridge University Press.
• Miller, R.K. (1971) Nonlinear Volterra Integral Equations. Menlo Park, CA: W. A. Benjamin.

Recommended Textbooks

• Brunner, H. (2017) Volterra Integral Equations: An Introduction to Theory and Applications. 2nd edn. Cambridge: Cambridge University Press.
• Gripenberg, G., Londen, S.-O. and Staffans, O. (1990) Volterra Integral and Functional Equations. Cambridge: Cambridge University Press.
• Miller, R.K. (1971) Nonlinear Volterra Integral Equations. Menlo Park, CA: W. A. Benjamin.