نبذة مختصرة : Introduction We ponder the following one-dimensional fourth-order parabolic equation Ut(x, t) + Uxxxx(x,t) = f(t), (x, t) E QT, (1) with the homogeneous boundary conditions ux(0, t) = ux (1, t) = Uxxx (0, t) = Uxxx (1, t) = 0, t∈ [0,T] (2) and extra measurements (3) T u(x, t) dt = g(x), x ∈ [0,1], 0 u(x*, t) = h(t), t∈ [0,T], (4) to approximate the unknown functions (f(t), u(x, t)). The system of equations ሺ1ሻ െ ሺ4ሻ, appears in the study of a wide range of problems derived from physics, chemistry, biology and engineering. The parabolic equations of the fourth order are used to create a balance between removing noise and preserving the edge and preventing blocky effects in image processing. We can also point to famous equations such as CahnHilliard and Kuramoto-Sivashinsky equations which are very important in describing instabilities in dynamical systems derived from physical and chemical phenomena. Investigations regarding the existence and uniqueness of the solution, regularity and presentation of strong and weak solutions for some parabolic equations of the fourth order have been done in several articles. Moreover, various numerical methods such as finite difference method, finite element method and spline approximation method have been used to solve these equations. MaterialandMethods The sufficient conditions of the existence and uniqueness of the solution for the considered inverse problem are established. A spectraltechniquebasedontheapplicationofoperationalmatricesof differentiation is applied for recovering the unknown functions. Numerical simulations are included to show the efficiency of the proposedmethod. Resultsanddiscussion By using the Fourier method and applying the assumptions of the problem, we prove the existence and uniqueness of the classical solution for the inverse problem expressed in equations (1) - (4). Then, we present an accurate and stable numerical algorithm to determine the unknown functions of the problem in the presence of accurate and contaminated input boundary data. It can be seen that by employing the proposed method, satisfactory results are obtained such that in the presence of the exact initial and boundary data, the unknown functions are excellently retrieved and regarding the noisy boundary data the obtained approximations deviate from the analytical solution almost proportional to the amount of introduced noise. Conclusion From this study, following concluding remarks were drawn. The solvability of the inverse problem is proved. An efficient numerical method is applied for approximating the unknown functions. In fact, the main problem is reduced to a linear system of algebraic equations which can be easily solved to get satisfactory findings. The technique can be developed for solving a broad class of initial/boundary value problems. We tested the numerical stability of the solution when dealing with noisy boundary conditions. It is seen that by applying the Tikhonov regularization method, robust and reliable solutions are achieved. [ABSTRACT FROM AUTHOR]
Processing Request
No Comments.