Simulation of 2D and 3D inverse source problems of nonlinear time-fractional wave equation by the meshless homogenization function method

Lin, Ji

doi:10.1007/s00366-021-01489-2

Cited by 19 publications

(9 citation statements)

References 41 publications

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“…, a 1 = 0, a 2 = 0, and ρ = ρ. (16) For the solutions of Equation (10), the first set (14) provides the following two cases: Case 1-1: Let 1 < 0; then, Equation (10) has the following solutions:…”

Section: Andmentioning

confidence: 99%

“…where θ = θ 1 x + θ 2 y + θ 3 t. Meanwhile, the third set (16) gives the solution of Equation ( 10) as follows:…”

Section: Andmentioning

confidence: 99%

“…The Hirota's [1], Riccati-Bernoulli sub-ODE [2], exp(−φ(ς))-expansion [3], perturbation [4][5][6][7], (G /G)-expansion [8,9], Jacobi elliptic function [10], sine-cosine [11,12], tanh-sech [13,14], etc, are some examples of analytical methods. While, a few numerical methods for solving fractional stochastic partial differential equations have been introduced including Galerkin finite element method [15], the meshless method [16,17], finite element method [18], implicit Euler method [19,20], the modified decomposition technique [21], and so on.…”

Section: Introductionmentioning

confidence: 99%

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The Impact of the Wiener Process on the Analytical Solutions of the Stochastic (2+1)-Dimensional Breaking Soliton Equation by Using Tanh–Coth Method

et al. 2022

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The stochastic (2+1)-dimensional breaking soliton equation (SBSE) is considered in this article, which is forced by the Wiener process. To attain the analytical stochastic solutions such as the polynomials, hyperbolic and trigonometric functions of the SBSE, we use the tanh–coth method. The results provided here extended earlier results. In addition, we utilize Matlab tools to plot 2D and 3D graphs of analytical stochastic solutions derived here to show the effect of the Wiener process on the solutions of the breaking soliton equation.

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Section: Andmentioning

confidence: 99%

“…where θ = θ 1 x + θ 2 y + θ 3 t. Meanwhile, the third set (16) gives the solution of Equation ( 10) as follows:…”

Section: Andmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Impact of the Wiener Process on the Analytical Solutions of the Stochastic (2+1)-Dimensional Breaking Soliton Equation by Using Tanh–Coth Method

et al. 2022

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“…To overcome the difficulties of conventional methods, in recent years, various numerical schemes [5][6][7] have been developed and successfully applied to Cauchy inverse problems, such as the boundary particle method (BPM), 8 the local radial basis function collocation method (LRBFCM), 9 the multiple/scale/direction Trefftz method (MSDTM), 10,11 the generalized finite difference method (GFDM), 12,13 the boundary knot method (BKM), [14][15][16] the method of fundamental solutions (MFS), 17,18 and the meshless homogenization function method. 19,20 Despite the successes of the application of these methods, there are still certain issues to be solved. For example, the dense matrices of the MFS and BKM significantly limit their applications in large-scale problems.…”

Section: Introductionmentioning

confidence: 99%

Localized MFS for three‐dimensional acoustic inverse problems on complicated domains

Chen

Wang

Cheng

et al. 2022

Int Journal of Mech Sys Dyn

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This paper proposes a semi‐analytical and local meshless collocation method, the localized method of fundamental solutions (LMFS), to address three‐dimensional (3D) acoustic inverse problems in complex domains. The proposed approach is a recently developed numerical scheme with the potential of being mathematically simple, numerically accurate, and requiring less computational time and storage. In LMFS, an overdetermined sparse linear system is constructed by using the known data at the nodes on the accessible boundary and by making the remaining nodes satisfy the governing equation. In the numerical procedure, the pseudoinverse of a matrix is solved via the truncated singular value decomposition, and thus the regularization techniques are not needed in solving the resulting linear system with a well‐conditioned matrix. Numerical experiments, involving complicated geometry and the high noise level, confirm the effectiveness and performance of the LMFS for solving 3D acoustic inverse problems.

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“…The phenomena of heat transfer may be found in many diffusion problems. Therefore, the investigation of partial differential equations such as the heat equation has much more application in real life [33,34]. The analysis for semi-analytical solutions for the problems of electrical circuits [35] has been performed using numerical approximation.…”

Section: Introductionmentioning

confidence: 99%

Semi-Analytical Solutions for Fuzzy Caputo–Fabrizio Fractional-Order Two-Dimensional Heat Equation

et al. 2021

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In the analysis in this article, we developed a scheme for the computation of a semi-analytical solution to a fuzzy fractional-order heat equation of two dimensions having some external diffusion source term. For this, we applied the Laplace transform along with decomposition techniques and the Adomian polynomial under the Caputo–Fabrizio fractional differential operator. Furthermore, for obtaining a semi-analytical series-type solution, the decomposition of the unknown quantity and its addition established the said solution. The obtained series solution was calculated and approached the approximate solution of the proposed equation. For the validation of our scheme, three different examples have been provided, and the solutions were calculated in fuzzy form. All the three illustrations simulated two different fractional orders between 0 and 1 for the upper and lower portions of the fuzzy solution. The said fractional operator is nonsingular and global due to the presence of the exponential function. It globalizes the dynamical behavior of the said equation, which is guaranteed for all types of fuzzy solution lying between 0 and 1 at any fractional order. The fuzziness is also included in the unknown quantity due to the fuzzy number providing the solution in fuzzy form, having upper and lower branches.

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Simulation of 2D and 3D inverse source problems of nonlinear time-fractional wave equation by the meshless homogenization function method

Cited by 19 publications

References 41 publications

The Impact of the Wiener Process on the Analytical Solutions of the Stochastic (2+1)-Dimensional Breaking Soliton Equation by Using Tanh–Coth Method

The Impact of the Wiener Process on the Analytical Solutions of the Stochastic (2+1)-Dimensional Breaking Soliton Equation by Using Tanh–Coth Method

Localized MFS for three‐dimensional acoustic inverse problems on complicated domains

Semi-Analytical Solutions for Fuzzy Caputo–Fabrizio Fractional-Order Two-Dimensional Heat Equation

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