Linear Partial Differential Equations of First Order as Bi-Dimensional Inverse Moments Problem

Author(s)
Maria Beatriz Pintarelli

ABSTRACT

We consider linear partial differential equations of first order

on a region . We will see that we can write the equation in partial derivatives as an Fredholm integral equation of the first kind and will solve this latter with the techniques of inverse problem moments. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on problem of moments.

We consider linear partial differential equations of first order

on a region . We will see that we can write the equation in partial derivatives as an Fredholm integral equation of the first kind and will solve this latter with the techniques of inverse problem moments. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on problem of moments.

Cite this paper

Pintarelli, M. (2015) Linear Partial Differential Equations of First Order as Bi-Dimensional Inverse Moments Problem.*Applied Mathematics*, **6**, 979-989. doi: 10.4236/am.2015.66090.

Pintarelli, M. (2015) Linear Partial Differential Equations of First Order as Bi-Dimensional Inverse Moments Problem.

References

[1] Akheizer, N.I. (1965) The Classical Moment Problem. Olivier and Boyd, Edinburgh.

[2] Akheizer, N.I. and Krein, M.G. (1962) Some Questions in the Theory of Moment. American Mathematical Society. Providence.

[3] Shohat, J.A. and Tamarkin, J.D. (1943) The Problem of Moments. Mathematical Surveys, American Mathematical Society, Providence.

http://dx.doi.org/10.1090/surv/001

[4] Ang, D.D., Gorenflo, R., Le, V.K. and Trong, D.D. (2002) Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction. Lectures Notes in Mathematics, Springer-Verlag, Berlin.

http://dx.doi.org/10.1007/b84019

[5] Pintarelli, M.B. and Vericat, F. (2012) Klein-Gordon Equation as a Bi-Dimensional Moment Problem. Far East Journal of Mathematical Sciences, 70, 201-225.

[6] Tikhonov, A. and Arsenine, V. (1976) Méthodes de résolution de problèmes mal posés. MIR, Moscow.

[7] Engl, H.W. and Groetsch, C.W. (1987) Inverse and Ill-Posed Problems. Academic Press, Boston.

[8] Pintarelli, M.B. and Vericat, F. (2008) Stability Theorem and Inversion Algorithm for a Generalized Moment Problem. Far East Journal of Mathematical Sciences, 30, 253-274.

[9] Pintarelli, M.B. and Vericat, F. (2011) Bi-Dimensional Inverse Moment Problems. Far East Journal of Mathematical Sciences, 54, 1-23.

[10] Talenti, G. (1987) Recovering a Function from a Finite Number of Moments. Inverse Problems, 3, 501-517.

http://dx.doi.org/10.1088/0266-5611/3/3/016

[11] Ames, W.F. (1992) Numerical Methods for Partial Differential Equations. Academic Press, New York.

[12] Lapidus, L. and Pinder, G.F. (1982) Numerical Solution of Partial Differential Equations in Science and Engineering. John Wiley and Sons, New York.

[13] Smith, G.D. (1985) Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford University Press, New York.

[14] Thomas, J.W. (1995) Numerical Partial Differential Equations: Finite Difference Methods. Springer-Verlag, New York.

http://dx.doi.org/10.1007/978-1-4899-7278-1

[1] Akheizer, N.I. (1965) The Classical Moment Problem. Olivier and Boyd, Edinburgh.

[2] Akheizer, N.I. and Krein, M.G. (1962) Some Questions in the Theory of Moment. American Mathematical Society. Providence.

[3] Shohat, J.A. and Tamarkin, J.D. (1943) The Problem of Moments. Mathematical Surveys, American Mathematical Society, Providence.

http://dx.doi.org/10.1090/surv/001

[4] Ang, D.D., Gorenflo, R., Le, V.K. and Trong, D.D. (2002) Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction. Lectures Notes in Mathematics, Springer-Verlag, Berlin.

http://dx.doi.org/10.1007/b84019

[5] Pintarelli, M.B. and Vericat, F. (2012) Klein-Gordon Equation as a Bi-Dimensional Moment Problem. Far East Journal of Mathematical Sciences, 70, 201-225.

[6] Tikhonov, A. and Arsenine, V. (1976) Méthodes de résolution de problèmes mal posés. MIR, Moscow.

[7] Engl, H.W. and Groetsch, C.W. (1987) Inverse and Ill-Posed Problems. Academic Press, Boston.

[8] Pintarelli, M.B. and Vericat, F. (2008) Stability Theorem and Inversion Algorithm for a Generalized Moment Problem. Far East Journal of Mathematical Sciences, 30, 253-274.

[9] Pintarelli, M.B. and Vericat, F. (2011) Bi-Dimensional Inverse Moment Problems. Far East Journal of Mathematical Sciences, 54, 1-23.

[10] Talenti, G. (1987) Recovering a Function from a Finite Number of Moments. Inverse Problems, 3, 501-517.

http://dx.doi.org/10.1088/0266-5611/3/3/016

[11] Ames, W.F. (1992) Numerical Methods for Partial Differential Equations. Academic Press, New York.

[12] Lapidus, L. and Pinder, G.F. (1982) Numerical Solution of Partial Differential Equations in Science and Engineering. John Wiley and Sons, New York.

[13] Smith, G.D. (1985) Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford University Press, New York.

[14] Thomas, J.W. (1995) Numerical Partial Differential Equations: Finite Difference Methods. Springer-Verlag, New York.

http://dx.doi.org/10.1007/978-1-4899-7278-1