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Fuzzy Mathematics in Economics and Engineering pp 165–184Cite as

Fuzzy Difference Equations

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Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 91))

Abstract

In this chapter we are concerned with solutions to the second order, linear, constant coefficient difference equation

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgU % caRiaadkeaaaa!3864!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$y\left( k+2 \right)+ay\left( k+1 \right)+by\left( k \right)=g\left( k \right)$$
(8.1)

for k = 0,1, 2,..., where a, b are constants with b > 0 and g(x) is continuous for x > O. The initial conditions are y(0) = γ0 and y(1) = γl. We will fuzzify the difference equation by considering fuzzy initial conditions EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgU % caRiaadkeaaaa!3864!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$y\left( 0 \right)=\overline{{{\gamma }_{0}}}$$, EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgU % caRiaadkeaaaa!3864!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$y\left( 1 \right)=\overline{{{\gamma }_{1}}}$$ for triangular fuzzy numbers EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgU % caRiaadkeaaaa!3864!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$\overline{{{\gamma }_{0}}}$$ and EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgU % caRiaadkeaaaa!3864!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$\overline{{{\gamma }_{1}}}$$.

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Bibliography

  1. J.J. Buckley, Th. Feuring and Y. Hayashi: Fuzzy Difference Equations: The Initial Value Problem. J. Advanced Computational Intelligence. To appear.

    Google Scholar 

  2. E.Y. Deeba and A. de Korvin: On a Fuzzy Difference Equation, IEEE Trans. Fuzzy Systems, 3 (1995), pp. 469–473.

    Article  Google Scholar 

  3. E.Y. Deeba and A. de Korvin: Analysis by Fuzzy Difference Equations of a Model of CO2 Level in the Blood, Applied Math. Letters, 12 (1999), pp. 33–40.

    Article  Google Scholar 

  4. E. Deeba, A. de Korvin and S. Xie: Techniques and Applications of Fuzzy Set Theory to Difference and Functional Equations and their Utilization in Modeling Diverse Systems, in: C.T. Leondes (Ed.), Fuzzy Theory Systems, Vol. 1, Academic Press, San Diego, CA., pp. 87–110.

    Google Scholar 

  5. The Fibonacci Quarterly, Fibonacci Association, Univ. Santa Clara, CA.

    Google Scholar 

  6. S. Goldberg: Introduction to Difference Equations, John Wiley and Sons, N. Y., 1958.

    Google Scholar 

  7. F.B. Hildebrand: Methods of Applied Mathematics, Prentice Hall, Englewood Cliffs, N.J., 1958.

    Google Scholar 

  8. F.B. Hildebrand: Finite—Difference Equations and Simulations, Prentice Hall, Englewood Cliffs, N.J., 1968.

    Google Scholar 

  9. W.G. Kelley and A.C. Peterson: Difference Equations: An Introduction with Applications, Second Edition, Harcourt/Academic Press, Burlington, MA, USA, 2000.

    Google Scholar 

  10. V. Lakashmikanthan and D. Trigante: Theory of Difference Equations, Academic Press, San Diego, CA, 1988.

    Google Scholar 

  11. K.S. Miller: Linear Difference Equations, W.A. Benjamin, N.Y., 1968.

    Google Scholar 

  12. P.A. Samuelson: Interactions Between the Multiplier Analysis and the Principle of Acceleration, Review of Economic Statistics, 21 (1939), pp. 75–78.

    Article  Google Scholar 

  13. C.E. Shannon and W. Weaver: The Mathematical Theory of Communication, Univ. of Illinois Press, Urbana, Ill., 1949.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematics Department, University of Alabama at Birmingham, 35294, Birmingham, AL, USA

    Professor James J. Buckley

  2. Department of Mathematics, Shahid Bahonar University, Kerman, Iran

    Professor Esfandiar Eslami

  3. Electrical Engineering and Computer Science, University of Siegen, Hölderlinstraße 3, 57068, Siegen, Germany

    Dr. Thomas Feuring

Authors
  1. Professor James J. Buckley
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  2. Professor Esfandiar Eslami
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  3. Dr. Thomas Feuring
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© 2002 Springer-Verlag Berlin Heidelberg

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Buckley, J.J., Eslami, E., Feuring, T. (2002). Fuzzy Difference Equations. In: Fuzzy Mathematics in Economics and Engineering. Studies in Fuzziness and Soft Computing, vol 91. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1795-9_8

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  • DOI: https://doi.org/10.1007/978-3-7908-1795-9_8

  • Publisher Name: Physica, Heidelberg

  • Print ISBN: 978-3-7908-2505-3

  • Online ISBN: 978-3-7908-1795-9

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