Fourier truncation method for the non-homogeneous time fractional backward heat conduction problem
03 Mar 2020-Inverse Problems in Science and Engineering (Taylor & Francis)-Vol. 28, Iss: 3, pp 402-426
TL;DR: In this article, the problem of determining the initial data for the backward non-homogeneous time fractional heat conduction problem by the Fourier truncation method is studied. But the exact solution is not known.
Abstract: This paper is devoted to the problem of determining the initial data for the backward non-homogeneous time fractional heat conduction problem by the Fourier truncation method. The exact solution fo...
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TL;DR: The terminal value problem for pseudo-parabolic equations with Riemann–Liouville fractional derivatives is considered, from a given final value and the existence (and regularity) of mild solutions is investigated.
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TL;DR: A modified version of quasi-boundary value method is applied to construct stable approximation problem for time-space fractional diffusion equations associated with the observed data measured in Hilbert scales to overcome the ill-posedness of the problem.
Abstract: This work is concerned with a mathematical study of backward problem for time-space fractional diffusion equations associated with the observed data measured in Hilbert scales. Transforming the original problem into an operator equation, we investigate the existence, the uniqueness and the instability for the problem. In order to overcome the ill-posedness of the problem, we apply a modified version of quasi-boundary value method to construct stable approximation problem. Using a Holder-type smoothness assumption of the exact solution it is shown that estimates achieve optimal rates of convergence in Hilbert scales both for an a-priori and for an a-posteriori parameter choice strategies.
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TL;DR: In this article, an interpolating element-free Galerkin (IEFG) method for the numerical study of the time-fractional diffusion equation, and then discuss the stability and convergence of the numerical solutions.
Abstract: This paper aims to present an interpolating element-free Galerkin (IEFG) method for the numerical study of the time-fractional diffusion equation, and then discuss the stability and convergence of the numerical solutions.,In the time-fractional diffusion equation, the time fractional derivatives are approximated by L1 method, and the shape functions are constructed by the interpolating moving least-squares (IMLS) method. The final system equations are obtained by using the Galerkin weak form. Because the shape functions have the interpolating property, the unknowns can be solved by the iterative method after imposing the essential boundary condition directly.,Both theoretical and numerical results show that the IEFG method for the time-fractional diffusion equation has high accuracy. The stability of the fully discrete scheme of the method on the time step is stable unconditionally with a high convergence rate.,This work will provide an interpolating meshless method to study the numerical solutions of the time-fractional diffusion equation using the IEFG method.
7 citations
TL;DR: In this article, the bi-Helmholtz equation with Cauchy conditions is nominated in a n-dimensional strip domain, and it is shown that this problem may be ill-posed in the sense of Hadamard.
Abstract: In this paper, the bi-Helmholtz equation with Cauchy conditions is nominated in a n-dimensional strip domain. It is shown that this Cauchy problem may be ill-posed in the sense of Hadamard. In orde...
4 citations
Cites methods from "Fourier truncation method for the n..."
...In ill-posed problems, a regularization parameter can be obtained by an a-periori and/or a-posteriori regularization parameter choice rules [29,30]....
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References
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Book•
01 Jan 1999
15,898 citations
"Fourier truncation method for the n..." refers result in this paper
...Recall also that (see [27]), for α > 0 and β > 0, the Mittag–Leffler function Eα,β is defined by...
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...Some basic results from fractional calculus Let us recall the definition and some results associated with Riemann–Liouville fractional integral, Caputo fractional derivative andMittag–Leffler function as given in [26,27]....
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Book•
02 Mar 2006
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
Abstract: 1. Preliminaries. 2. Fractional Integrals and Fractional Derivatives. 3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems. 4. Methods for Explicitly solving Fractional Differential Equations. 5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations. 6. Partial Fractional Differential Equations. 7. Sequential Linear Differential Equations of Fractional Order. 8. Further Applications of Fractional Models. Bibliography Subject Index
11,492 citations
"Fourier truncation method for the n..." refers background or result in this paper
...We may recall that Riemann–Liouville fractional integral and Caputo fractional derivative can be defined for any α > 0 and for t ∈ [0,∞) for appropriate functions (see [26])....
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...Some basic results from fractional calculus Let us recall the definition and some results associated with Riemann–Liouville fractional integral, Caputo fractional derivative andMittag–Leffler function as given in [26,27]....
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TL;DR: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.
7,412 citations
"Fourier truncation method for the n..." refers background in this paper
...Introduction Currently, a lot of research activities on the time fractional diffusion equations has been taking place due to its importance in several areas of science and engineering such as for describing the memory as well as hereditary properties for super diffusion and subdiffusion phenomena in the theory of plasma turbulence [1,2], randomwalks [3,4], viscoelastic material, biological systems [5] and in various other physical models [6]....
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Book•
01 Jan 1964
TL;DR: The real and complex number system as discussed by the authors is a real number system where the real number is defined by a real function and the complex number is represented by a complex field of functions.
Abstract: Chapter 1: The Real and Complex Number Systems Introduction Ordered Sets Fields The Real Field The Extended Real Number System The Complex Field Euclidean Spaces Appendix Exercises Chapter 2: Basic Topology Finite, Countable, and Uncountable Sets Metric Spaces Compact Sets Perfect Sets Connected Sets Exercises Chapter 3: Numerical Sequences and Series Convergent Sequences Subsequences Cauchy Sequences Upper and Lower Limits Some Special Sequences Series Series of Nonnegative Terms The Number e The Root and Ratio Tests Power Series Summation by Parts Absolute Convergence Addition and Multiplication of Series Rearrangements Exercises Chapter 4: Continuity Limits of Functions Continuous Functions Continuity and Compactness Continuity and Connectedness Discontinuities Monotonic Functions Infinite Limits and Limits at Infinity Exercises Chapter 5: Differentiation The Derivative of a Real Function Mean Value Theorems The Continuity of Derivatives L'Hospital's Rule Derivatives of Higher-Order Taylor's Theorem Differentiation of Vector-valued Functions Exercises Chapter 6: The Riemann-Stieltjes Integral Definition and Existence of the Integral Properties of the Integral Integration and Differentiation Integration of Vector-valued Functions Rectifiable Curves Exercises Chapter 7: Sequences and Series of Functions Discussion of Main Problem Uniform Convergence Uniform Convergence and Continuity Uniform Convergence and Integration Uniform Convergence and Differentiation Equicontinuous Families of Functions The Stone-Weierstrass Theorem Exercises Chapter 8: Some Special Functions Power Series The Exponential and Logarithmic Functions The Trigonometric Functions The Algebraic Completeness of the Complex Field Fourier Series The Gamma Function Exercises Chapter 9: Functions of Several Variables Linear Transformations Differentiation The Contraction Principle The Inverse Function Theorem The Implicit Function Theorem The Rank Theorem Determinants Derivatives of Higher Order Differentiation of Integrals Exercises Chapter 10: Integration of Differential Forms Integration Primitive Mappings Partitions of Unity Change of Variables Differential Forms Simplexes and Chains Stokes' Theorem Closed Forms and Exact Forms Vector Analysis Exercises Chapter 11: The Lebesgue Theory Set Functions Construction of the Lebesgue Measure Measure Spaces Measurable Functions Simple Functions Integration Comparison with the Riemann Integral Integration of Complex Functions Functions of Class L2 Exercises Bibliography List of Special Symbols Index
6,681 citations
Book•
01 Jan 1998
TL;DR: Inverse problems and regularization of the Cauchy problem have been studied in this article, with a focus on the uniqueness and stability of the regularization process of the problem.
Abstract: Inverse Problems- Ill-Posed Problems and Regularization- Uniqueness and Stability in the Cauchy Problem- Elliptic Equations: Single Boundary Measurements- Elliptic Equations: Many Boundary Measurements- Scattering Problems- Integral Geometry and Tomography- Hyperbolic Problems- Inverse parabolic problems- Some Numerical Methods
1,709 citations
"Fourier truncation method for the n..." refers background in this paper
...For detailed study on regularization of partial differential equation and for regularization of linear ill-posed operator equations, one may refer Isakov [16] and Nair [17], respectively....
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