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A Fundamental Theorem on Initial Value Problems by Using the Theory of Reproducing Kernels

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Abstract

We introduce a new method for solving general initial value problems by using the theory of reproducing kernels. The results are depending on the specific structure of each problem. Here, we give the general principle of the method and illustrate it with simple prototype examples. On the basis of the process, we have certain integral transforms, which are generated by each specific initial value problem, and need to be analysed. In view of this, we shall establish the basic relations among initial value problems for linear operator equations, eigenvalues and eigenfunctions in the related operator equations, integral transforms and associated reproducing kernels. Within this process, we will realize a general theory for operator equations and incorporate a time dependence in view to consider an associated regularization method.

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References

  1. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  2. Castro, L.P., Chen, Q., Saitoh, Q.: Source inversion of heat conduction from a finite number of observation data. Appl. Anal. 89, 801–813 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Castro, L.P., Fujiwara, H., Rodrigues, M.M., Saitoh, S., Tuan, V.K.: Aveiro discretization method in mathematics: a new discretization principle. In: Pardalos, P., Rassias, T.M. (eds.) Mathematics Without Boundaries: Surveys in Pure Mathematics, p. 52. Springer, New York (2014). http://www.springer.com/mathematics/analysis/book/978-1-4939-1105-9

  4. Castro, L.P., Fujiwara, H., Qian, T., Saitoh, S.: How to catch smoothing properties and analyticity of functions by computers? In: Pardalos, P., Rassias, T.M. (eds.) Mathematics Without Boundaries: Surveys in Interdisciplinary Research, p. 15. Springer, New York (2014). http://www.springer.com/mathematics/analysis/book/978-1-4939-1123-3

  5. Castro, L.P., Itou, H., Saitoh, S.: Numerical solutions of linear singular integral equations by means of Tikhonov regularization and reproducing kernels. Houst. J. Math. 38(4), 1261–1276 (2012)

    MATH  MathSciNet  Google Scholar 

  6. Castro, L.P., Saitoh, S.: Optimal and approximate solutions of singular integral equations by means of reproducing kernels. Complex Anal. Oper. Theory 7, 1839–1851 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gradshlein, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Elsevier/Academic Press, Amsterdam (2007)

    Google Scholar 

  8. Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series. More Special Functions, vol. 3. Gordon and Breach Publisher, New York (1990)

  9. Polyanin, A.D.: Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman & Hall/CRC, Boca Raton (2002)

    MATH  Google Scholar 

  10. Polyanin, A.D., Zaitsev, V.F.: Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press, Boca Raton (2003)

    MATH  Google Scholar 

  11. Polyanin, A.D., Manzhirov, A.V.: Handbook of Integral Equations. CRC Press, Boca Raton (2008)

    Book  MATH  Google Scholar 

  12. Saitoh, S.: Hilbert spaces induced by Hilbert space valued functions. Proc. Am. Math. Soc. 89, 74–78 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  13. Saitoh, S.: Integral Transforms, Reproducing Kernels and their Applications. Pitman Research Notes in Mathematics Series, vol. 369. Addison Wesley Longman, Harlow (1997)

  14. Saitoh, S.: Theory of reproducing kernels: applications to approximate solutions of bounded linear operator functions on Hilbert spaces. Am. Math. Soc. Transl. Ser. 230, 107–134 (2010)

    MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors were supported in part by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (“FCT—Fundação para a Ciência e a Tecnologia”) within project PEst-OE/MAT/UI4106/2014. The third author is supported in part by the Grant-in-Aid for the Scientific Research (C)(2)(No. 24540113)

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Authors and Affiliations

  1. Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 , Aveiro, Portugal

    L. P. Castro, M. M. Rodrigues & S. Saitoh

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  1. L. P. Castro
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  2. M. M. Rodrigues
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  3. S. Saitoh
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Correspondence to L. P. Castro.

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Communicated by Daniel Aron Alpay.

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Castro, L.P., Rodrigues, M.M. & Saitoh, S. A Fundamental Theorem on Initial Value Problems by Using the Theory of Reproducing Kernels. Complex Anal. Oper. Theory 9, 87–98 (2015). https://doi.org/10.1007/s11785-014-0375-1

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  • DOI: https://doi.org/10.1007/s11785-014-0375-1

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