Application of the Ritz–Galerkin method for recovering the spacewise-coefficients in the wave equation

Rashedi, Kamal; Adibi, Hojatollah; Dehghan, Mehdi

doi:10.1016/j.camwa.2013.04.005

Cited by 17 publications

(17 citation statements)

References 38 publications

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“…The numerical methods for solving the wave speed identification problems have been examined by many researchers, for example Refs. [10][11][12][13][14][15]. Note that we do not need the data from the initial displacement and initial velocity, which are drastically different from that in the above numerical methods.…”

Section: Solving the Inverse Problems Of Wave Equation By A Boundary mentioning

confidence: 99%

Solving the Inverse Problems of Wave Equation by a Boundary Functional Method

Liu¹,

Chen²

2017

JSOE

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Abstract:The inverse problems of wave equation to recover unknown space-time dependent functions of wave speed and wave source are solved in this paper, without needing of initial conditions and no internal measurement of data being required. After a homogenization technique, a sequence of spatial boundary functions at least the fourth-order polynomials are derived, which satisfy the homogeneous boundary conditions. The boundary functions and the zero element constitute a linear space, and then a new boundary functional is proved in the linear space, of which the energy is preserved for each dynamic energetic boundary function. The linear systems and iterative algorithms used to recover unknown wave speed and wave source functions with the dynamic energetic boundary functions as bases are developed, which converge fast at each time step. The input data are parsimonious, merely the measured boundary strains and the boundary values and slopes of unknown functions to be recovered. The accuracy and robustness of present methods are confirmed by comparing exact solutions with estimated results under large noises up to 20%.

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Section: Solving the Inverse Problems Of Wave Equation By A Boundary mentioning

confidence: 99%

Solving the Inverse Problems of Wave Equation by a Boundary Functional Method

Liu¹,

Chen²

2017

JSOE

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“…Several partial differential equations are numerically solved by Ritz-Galerkin method, but using of the appropriate satisfier function in the Ritz-Galerkin method is taken into consideration recently, see for instance [14][15][16][17][18][19][20]. The satisfier function fulfills all the problem conditions.…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional Bernoulli wavelets with satisfier function in the Ritz–Galerkin method for the time fractional diffusion-wave equation with damping

Barikbin

2017

Math Sci

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In this paper, the two-dimensional Bernoulli wavelets (BWs) with Ritz-Galerkin method are applied for the numerical solution of the time fractional diffusionwave equation. In this way, a satisfier function which satisfies all the initial and boundary conditions is derived. The two-dimensional BWs and Ritz-Galerkin method with satisfier function are used to transform the problem under consideration into a linear system of algebraic equations. The proposed scheme is applied for numerical solution of some examples. It has high accuracy in computation that leads to obtaining the exact solutions in some cases.

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“…It is clear that f n is unbounded while the whole boundary conditions tend to zero as n −→ ∞ which shows the instability in retrieving the source function f (t). Therefore dealing with this problem requires employing appropriate procedures to produce stable solutions [17,20].…”

Section: Introductionmentioning

confidence: 99%

“…Prior to our work, the Ritz-Galerkin method, known as a domain Galerkin technique, dealt with several linear and nonlinear partial differential equations, see for instance [9,20,21,27]. Briefly stated, we solved the problem for not only the standard initial and boundary conditions [9,21], but also the nonlocal boundary conditions [20] through an auxiliary function called "satisfier function". In conclusion, the large sets of collocation points are not needed for applying the supplemented boundary conditions which naturally leads to a system of algebraic equations of smaller size and hence reduces the computation time.…”

Section: Introductionmentioning

confidence: 99%

Stable recovery of the time-dependent source term from one measurement for the wave equation

Rashedi

Sini

2015

Inverse Problems

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In this work, we discuss the inverse problem which consists of the determination of an unknown time-dependent force function from one time-dependent measurement collected in any space point for the one-dimensional wave equation. This problem is motivated by the question of estimating the timedependent body force which needs to be exerted on a given string to reach a desired shape at the final time. We prove its unique solvability using as data a linear combination of displacement and flux measured at one arbitrary fixed point of the string. We also derive a conditional Hölder stability estimate of this inverse problem. The numerical solution of the problem is investigated by means of the Ritz-Galerkin technique along with the application of the satisfier function to obtain cost-effective and stable results. Some numerical examples are provided to show the performance of the proposed scheme.

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Application of the Ritz–Galerkin method for recovering the spacewise-coefficients in the wave equation

Cited by 17 publications

References 38 publications

Solving the Inverse Problems of Wave Equation by a Boundary Functional Method

Solving the Inverse Problems of Wave Equation by a Boundary Functional Method

Two-dimensional Bernoulli wavelets with satisfier function in the Ritz–Galerkin method for the time fractional diffusion-wave equation with damping

Stable recovery of the time-dependent source term from one measurement for the wave equation

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