A Final Value Problem for the Homogeneous Space Fractional Damped Wave Equation with Gaussian White Noise
Main Article Content
Abstract
In this work, we consider the problem for the homogeneous space fractional damped wave equation with Gaussian white noise. As commonly acknowledged, the problem is severely ill-posed according to Hadamard's sense. Consequently, we propose the Fourier truncation method to regularize the problem. With different assumptions on the exact solution, the estimation of the expectation of the error between the regularized solution and the exact solution in - norm is obtained. Finally, we provide an example to illustrate our theoretically obtained results.
Keywords:
Space fractional equation, damped wave equation, truncation method, regularization, Gaussian white noise. MSC: 35L05, 47J06, 47H10, 60G15.
References
M. Modanli, S. T. Abdulazeez, A. M. Husien, A New Approach for Pseudo Hyperbolic Partial Differential Equations with Nonlocal Consitions Using Laplace Adomian Decomosition Method, Applied Mathematics- A Journal of Chinese universities,Vol. 39, 2024, pp. 750-758.
S. Mesloub, H. A. H. Gadain, L. Kasmi, On the Well Posedness of A Mathematical Model for A Singular Nonlinear Fraction Pseudo - Hyperbolic System with Nonlocal Boundary Conditions and Frictional Damping Terms, AIMS Mathematics,Vol. 9, Iss. 2, 2024.
D. T. Anh, H. Michihisa, Study of Semi - linear σ - evolution Equations with Frictional and Visco - Elastic Damping, AIMS, Vol. 19, Iss. 3, 2020, pp. 1581-1608.
D. T. Anh, D. V. Duong, N. D. Anh, On Asymptotic Properties of Solutions to σ Evolution Equations with General Double Damping, Journal of Mathematical Analysis and Applications, Vol. 536, 2024.
W. Chen, M. D'Abbicco, G. Girardi, Global Small Data Solutions for Semilinear Waves with Two Dissipative Terms, Annali di Matematica Pura ed Applicata, Vol. 201, 2022, pp. 529-560.
N. H. Can, N. H. Tuan, D. O'Regan, V. V. Au, On a Final Value Problem for A Class of Nonlinear Hyperbolic Equations with Damping Term, Evolution Equations and Control Theory, Vol 1, No. 1, 2021.
B. Kaltenbacher, W. Rundell, Inverse Problems for Fractional Partial Differential Equations, American Mathematical Society, 2023.
N. D. Phuong, N. H. Tuan, Z. Hammouch, R. Sakthivel, On a Pseudo - parabolic Equations with A Non - Local Term of the Kirchhoff Type with Random Gaussian White Noise, Chaos, Solitons and Fractals, Vol. 145, 2021.
M. Q. Vinh, E. Nane, D. O'Regan, N. H. Tuan, Terminal Value Problem for Nonlinear Parabolic Equation with Gaussian White Noise, Electronic Research Archive, Vol. 30, No. 4, 2022, pp.1374-1413.
N. D. Phuong, E. Nane, O. Nikan, N. A. Tuan, Approximation of the Initial Value for Damped Nonlinear Hyperbolic Equations with Random Gaussian White Noise on the Measurements, AIMS Mathematics, Vol. 7, No. 7, 2022, pp. 12620-12634.
Z. Song, H. Di. Instability Analysis and Regularization Approximation to the Forward/Backward Problems for Fractional Damped Wave Equations with Random Noise, Applied Numerical Mathematics, Vol. 199, 2024,
pp. 177-212.
Yu. V. Egorov, M. A. Shubin, Foundations of the Classical Theory of Partial Differential Equations, Springer Berlin, 2013.
G. F. Roach, I. G. Stratis, A. N. Yannacopoulos, Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics, Princeton University Press, 2012.
L. Cavalier; Nonparametric Statistical Inverse Problems, Inverse Problem, Vol. 19, No. 24, 2008, pp. 034-004.
S. Mesloub, H. A. H. Gadain, L. Kasmi, On the Well Posedness of A Mathematical Model for A Singular Nonlinear Fraction Pseudo - Hyperbolic System with Nonlocal Boundary Conditions and Frictional Damping Terms, AIMS Mathematics,Vol. 9, Iss. 2, 2024.
D. T. Anh, H. Michihisa, Study of Semi - linear σ - evolution Equations with Frictional and Visco - Elastic Damping, AIMS, Vol. 19, Iss. 3, 2020, pp. 1581-1608.
D. T. Anh, D. V. Duong, N. D. Anh, On Asymptotic Properties of Solutions to σ Evolution Equations with General Double Damping, Journal of Mathematical Analysis and Applications, Vol. 536, 2024.
W. Chen, M. D'Abbicco, G. Girardi, Global Small Data Solutions for Semilinear Waves with Two Dissipative Terms, Annali di Matematica Pura ed Applicata, Vol. 201, 2022, pp. 529-560.
N. H. Can, N. H. Tuan, D. O'Regan, V. V. Au, On a Final Value Problem for A Class of Nonlinear Hyperbolic Equations with Damping Term, Evolution Equations and Control Theory, Vol 1, No. 1, 2021.
B. Kaltenbacher, W. Rundell, Inverse Problems for Fractional Partial Differential Equations, American Mathematical Society, 2023.
N. D. Phuong, N. H. Tuan, Z. Hammouch, R. Sakthivel, On a Pseudo - parabolic Equations with A Non - Local Term of the Kirchhoff Type with Random Gaussian White Noise, Chaos, Solitons and Fractals, Vol. 145, 2021.
M. Q. Vinh, E. Nane, D. O'Regan, N. H. Tuan, Terminal Value Problem for Nonlinear Parabolic Equation with Gaussian White Noise, Electronic Research Archive, Vol. 30, No. 4, 2022, pp.1374-1413.
N. D. Phuong, E. Nane, O. Nikan, N. A. Tuan, Approximation of the Initial Value for Damped Nonlinear Hyperbolic Equations with Random Gaussian White Noise on the Measurements, AIMS Mathematics, Vol. 7, No. 7, 2022, pp. 12620-12634.
Z. Song, H. Di. Instability Analysis and Regularization Approximation to the Forward/Backward Problems for Fractional Damped Wave Equations with Random Noise, Applied Numerical Mathematics, Vol. 199, 2024,
pp. 177-212.
Yu. V. Egorov, M. A. Shubin, Foundations of the Classical Theory of Partial Differential Equations, Springer Berlin, 2013.
G. F. Roach, I. G. Stratis, A. N. Yannacopoulos, Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics, Princeton University Press, 2012.
L. Cavalier; Nonparametric Statistical Inverse Problems, Inverse Problem, Vol. 19, No. 24, 2008, pp. 034-004.