Abstract
This paper provides a new numerical solution to partial differential equations, where the initial and boundary conditions are given by interactive fuzzy numbers. The interactivity considered here is the one obtained from the joint possibility distribution I, which is associated with a parameterized family of joint possibility distributions. The proposed method is given by the finite difference method, adapted for arithmetic operations of I-interactive fuzzy numbers. In addition to the proposed solution, a comparison with a numerical solution given by the fuzzy standard arithmetic, which is also known as non-interactive arithmetic, is presented. This paper shows that the numerical solution via interactive arithmetic is more specific than the one via non-interactive arithmetic, which means that the numerical solution via I-interactive arithmetic propagates less uncertainty than the numerical solution via standard arithmetic. The proposed method can be applied in any fuzzy partial and ordinary differential equation. In order to illustrate the results, an application to the heat equation is presented.
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References
LeVeque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, Philadelphia (1955)
Xiu, D., Karniadakis, G.: A new stochastic approach to transient heat conduction modeling with uncertainty. Int. J. Heat Mass Transfer 46, 4681–4693 (2003)
Wasques, V.F., Esmi, E., Barros, L.C., Bede, B.: Comparison between numerical solutions of fuzzy initial-value problems via interactive and standard arithmetics. In: Kearfott, R.B., Batyrshin, I., Reformat, M., Ceberio, M., Kreinovich, V. (eds.) IFSA/NAFIPS 2019 2019. AISC, vol. 1000, pp. 704–715. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-21920-8_62
Pinto, N.J.B., Wasques, V.F., Esmi, E., Barros, L.C.: Least squares method with interactive fuzzy coefficient: application on longitudinal data. In: Barreto, G.A., Coelho, R. (eds.) NAFIPS 2018. CCIS, vol. 831, pp. 132–143. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-95312-0_12
Sussner, P., Esmi, E., Barros, L.C.: Controlling the width of the sum of interactive fuzzy numbers with applications to fuzzy initial value problems. In: IEEE International Conference on Fuzzy Systems, pp. 1453–1460 (2016). https://doi.org/10.1109/FUZZ-IEEE.2016.7737860
Esmi, E., Sussner, P., Ignácio, G.B.D., Barros, L.C.: A parametrized sum of fuzzy numbers with applications to fuzzy initial value problems. Fuzzy Sets Syst. 331, 85–104 (2018)
Wasques, V.F., Esmi, E., Barros, L.C., Sussner, P.: The generalized fuzzy derivative is interactive. Inf. Sci. 519, 93–109 (2020)
Esmi, E., Wasques, V.F., Barros, L.C.: Addition and subtraction of interactive fuzzy numbers via family of joint possibility distributions. Fuzzy Sets Syst. (2021)
Wasques, V.F., Esmi, E., Barros, L.C., Sussner, P.: Numerical solution for fuzzy initial value problems via interactive arithmetic: application to chemical reactions. Int. J. Comput. Intell. Syst. 13(1), 1517–1529 (2020)
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Wasques, V.F. (2022). Comparison Between Numerical Solutions of Fuzzy Partial Differential Equations via Interactive and Non-interactive Arithmetics: Application to the Heat Equation. In: Kahraman, C., Cebi, S., Cevik Onar, S., Oztaysi, B., Tolga, A.C., Sari, I.U. (eds) Intelligent and Fuzzy Techniques for Emerging Conditions and Digital Transformation. INFUS 2021. Lecture Notes in Networks and Systems, vol 307. Springer, Cham. https://doi.org/10.1007/978-3-030-85626-7_96
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DOI: https://doi.org/10.1007/978-3-030-85626-7_96
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