Calculating heat and wave propagation from lateral Cauchy data

  • R. Chapko Ivan Franko Nat. Univ. Lviv, Ukraine
  • B. T. Johansson Linköping Univ., Sweden
Keywords: parabolic and hyperbolic Cauchy problem; 2- and 3-dimensional doubly connected domains; single-layer potential; boundary integral equations; method of fundamental solutions; trigonometric quadrature method; Tikhonov regularization.


UDC 519.6

We give an overview of recent methods based on semi-discretisation in time for the inverse ill-posed problem of calculating the solution of evolution equations from time-like Cauchy data. Specifically, the function value and normal derivative are given on a portion of the lateral boundary of a space-time cylinder and the corresponding data is to be generated on the remaining lateral part of the cylinder for either the heat or wave equation. The semi-discretisation in time constitutes of applying the Laguerre transform or the Rothe method (finite difference approximation), and has the feature that the similar sequence of elliptic problems is obtained for both the heat and wave equation, only the values of certain parameters change. The elliptic equations are solved numerically by either a boundary integral approach involving the Nystreom method or a method of fundamental solutions (MFS). Theoretical properties are stated together with discretisation strategies in space. Systems of linear equations are obtained for finding values of densities or coefficients. Tikhonov regularization is incorporated for the stable solution of the linear equations. Numerical results included show that the proposed strategies give good accuracy with an economical computational cost.


M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publ., New York (1972).

C. J. S.Alves, On the choice of source points in the method of fundamental solutions, Eng. Anal. Bound. Elem., 33, 1348 – 1361 (2009), DOI:

M. Bellassoued, M. Yamamoto, Carleman estimates and applications to inverse problems for hyperbolic systems, Springer-Verlag, Tokyo (2017), DOI:

I. Borachok, R. Chapko, B. T. Johansson, A method of fundamental solutions for heat and wave propagation from lateral Cauchy data, Numer. Algorithms, (2021). . DOI:

I. Borachok, R. Chapko, B. T. Johansson, A method of fundamental solutions with time-discretisation for wave motion from lateral Cauchy data, J. Sci. Comput. (to appear).

Y. H. Cao, L. H. Kuo, Hybrid method of space-time and Houbolt methods for solving linear time-dependent problems, Eng. Anal. Bound. Elem., 128, 58 – 65 (2021), DOI:

R. Chapko, B. T. Johansson, A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems, Appl. Numer. Math., 129, 104 – 119 (2018), DOI:

R. Chapko, B. T. Johansson, Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations, J. Eng. Math., 103, 23 – 37 (2017), DOI:

R. Chapko, B. T. Johansson, On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach, Inverse Probl. Imaging, 6, 25 – 38 (2012), DOI:

R. Chapko, B. T. Johansson, Y. Muzychuk, A. Hlova, Wave, propagation from lateral Cauchy data using a boundary element method, Wave Motion, 91 (2019), DOI:

R. Chapko, B. T. Johansson, Y. Savka, On the use of an integral equation approach for the numerical solution of a Cauchy problem for Laplace equation in a doubly connected planar domain, Inverse Probl. Sci. Eng., 22, 130 – 149 (2014), DOI:

R. Chapko, R. Kress, Rothe’s method for the heat equation and boundary integral equations, J. Integral Equat. and Appl., 9, 47 – 69 (1997), DOI:

R. Chapko, R. Kress, On the numerical solution of initial boundary value problems by the Laguerre transformation and boundary integral equations, Ser. Math. Anal. and Appl., vol. 2, Integral and Integrodifferential Equations: Theory, Methods and Applications, Gordon and Breach Sci. Publ., Amsterdam, (2000), p. 55 – 69 .

G. Fairweather, A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9, no. 1-2, 69 – 95 (1998), DOI:

H`ao, Dinh Nho, Methods for inverse heat conduction problems, Peter Lang, Frankfurt am Main (1998), DOI:

A. Hasanov Hasanoğlu, V. G. Romanov, Introduction to inverse problems for differential equations, Springer, Cham (2017), DOI:

J. C. Houbolt, A recurrence matrix solution for the dynamic response of elastic aircraft, J. Aeronaut. Sci., 17, 540 – 550 (1950). DOI:

V. Isakov, Inverse problems for partial differential equations, 3rd, Springer, Cham (2017), DOI:

V. M. Kaĭstrenko, The Cauchy problem for a second order hyperbolic equation with data on a time-like surface, Sibirsk. Mat. Zh., 16, 395 – 398 (1975); English translation: Sib. Math. J., 16, 306 – 308 (1975).

A. Karageorghis, D. Lesnic, L. Marin, A survey of applications of the MFS to inverse problems, Inverse Probl. Sci. Eng., 19, 309 – 336 (2011), DOI:

M. V.Klibanov, Carleman estimates for the regularization of ill-posed Cauchy problems, Appl. Numer. Math., 94, 46 – 74 (2015), DOI:

M. M. Lavren´tev, V. G. Romanov, S. P. Shishatski˘ı, Ill-posed problems of mathematical physics and analysis, Amer. Math. Soc., Providence, RI (1986). DOI:

How to Cite
Chapko, R., and B. T. Johansson. “Calculating Heat and Wave Propagation from Lateral Cauchy Data”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 2, Feb. 2022, pp. 274 -85, doi:10.37863/umzh.v74i2.6880.
Research articles