D. Uma, H. Jafari, S. Raja Balachandar, S. G. Venkatesh, S. Vaidyanathan
In this paper, approximate solutions for stochastic Fitzhugh–Nagumo partial differential equations are obtained using two‐dimensional shifted Legendre polynomial (2DSLP) approximation. The problem's suitability and solvability are confirmed. The convergence analysis for the proposed methodology and the error analysis in the norm are carried out. Using Maple software, an algorithm is created and implemented to arrive at the numerical solution. The solution thus obtained is compared with the exact solution and the solution obtained using the explicit order RK1.5 method.
本文利用二维移位 Legendre 多项式（2DSLP）近似法求得了随机 Fitzhugh-Nagumo 偏微分方程的近似解。证实了问题的适用性和可解性。对提出的方法进行了收敛性分析和规范误差分析。使用 Maple 软件创建并实施了一种算法，以获得数值解。将获得的解与精确解以及使用显式阶 RK1.5 方法获得的解进行了比较。
{"title":"An approximate solution for stochastic Fitzhugh–Nagumo partial differential equations arising in neurobiology models","authors":"D. Uma, H. Jafari, S. Raja Balachandar, S. G. Venkatesh, S. Vaidyanathan","doi":"10.1002/mma.10471","DOIUrl":"https://doi.org/10.1002/mma.10471","url":null,"abstract":"In this paper, approximate solutions for stochastic Fitzhugh–Nagumo partial differential equations are obtained using two‐dimensional shifted Legendre polynomial (2DSLP) approximation. The problem's suitability and solvability are confirmed. The convergence analysis for the proposed methodology and the error analysis in the norm are carried out. Using Maple software, an algorithm is created and implemented to arrive at the numerical solution. The solution thus obtained is compared with the exact solution and the solution obtained using the explicit order RK1.5 method.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this paper is to investigate similarity and consimilarity of hyper‐dual generalized quaternions and their matrices. For this purpose, we give different conjugates according to the generalized quaternionic units . We present ‐consimilarity of hyper‐dual generalized quaternions and their matrices except hyper‐dual ‐quaternions. For the generalization consisting of hyper‐dual coefficients quaternion and split quaternion, we search ‐consimilarity and ‐consimilarity with the help of ‐conjugate and ‐conjugate. We also give ‐coneigenvalues and ‐coneigenvectors of the matrices of these generalizations. In addition, we examine right coneigenvalue problem in generalized quaternion matrices for real and split quaternions. The complex matrix representation obtained through the complex adjoint matrix representation of this generalization is introduced, and its properties are presented. Besides, we give algebraic methods for the concept of right coneigenvalues and coneigenvectors for matrices, which are the generalization of real quaternion and split quaternion.
{"title":"Similarity and consimilarity of hyper‐dual generalized quaternions","authors":"Yasemin Alagöz, Gözde Özyurt","doi":"10.1002/mma.10488","DOIUrl":"https://doi.org/10.1002/mma.10488","url":null,"abstract":"The aim of this paper is to investigate similarity and consimilarity of hyper‐dual generalized quaternions and their matrices. For this purpose, we give different conjugates according to the generalized quaternionic units . We present ‐consimilarity of hyper‐dual generalized quaternions and their matrices except hyper‐dual ‐quaternions. For the generalization consisting of hyper‐dual coefficients quaternion and split quaternion, we search ‐consimilarity and ‐consimilarity with the help of ‐conjugate and ‐conjugate. We also give ‐coneigenvalues and ‐coneigenvectors of the matrices of these generalizations. In addition, we examine right coneigenvalue problem in generalized quaternion matrices for real and split quaternions. The complex matrix representation obtained through the complex adjoint matrix representation of this generalization is introduced, and its properties are presented. Besides, we give algebraic methods for the concept of right coneigenvalues and coneigenvectors for matrices, which are the generalization of real quaternion and split quaternion.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The uniform exponential stabilities (UESs) of two hybrid control systems comprised of a wave equation and a second‐order ordinary differential equation are investigated in this study. Linear feedback law and local viscosity are considered, as are nonlinear feedback law and internal anti‐damping. The hybrid system is first reduced to a first‐order port‐Hamiltonian system with dynamical boundary conditions, and the resulting system is discretized using the average central‐difference scheme. Second, the UES of the discrete system is obtained without prior knowledge of the exponential stability of the continuous system. The frequency domain characterization of UES for a family of contractive semigroups and the discrete multiplier approach are used to validate the main conclusions. Finally, the Trotter–Kato theorem is used to perform a convergence study on the numerical approximation approach. Most notably, the exponential stability of the continuous system is derived by the convergence of energy and UES, which is a novel approach to studying the exponential stability of some complex systems. Numerical simulation is used to validate the effectiveness of the numerical approximating strategy.
{"title":"Uniform exponential stability approximations of semi‐discretization schemes for two hybrid systems","authors":"Lu Zhang, Fu Zheng, Sizhe Wang, Zhongjie Han","doi":"10.1002/mma.10484","DOIUrl":"https://doi.org/10.1002/mma.10484","url":null,"abstract":"The uniform exponential stabilities (UESs) of two hybrid control systems comprised of a wave equation and a second‐order ordinary differential equation are investigated in this study. Linear feedback law and local viscosity are considered, as are nonlinear feedback law and internal anti‐damping. The hybrid system is first reduced to a first‐order port‐Hamiltonian system with dynamical boundary conditions, and the resulting system is discretized using the average central‐difference scheme. Second, the UES of the discrete system is obtained without prior knowledge of the exponential stability of the continuous system. The frequency domain characterization of UES for a family of contractive semigroups and the discrete multiplier approach are used to validate the main conclusions. Finally, the Trotter–Kato theorem is used to perform a convergence study on the numerical approximation approach. Most notably, the exponential stability of the continuous system is derived by the convergence of energy and UES, which is a novel approach to studying the exponential stability of some complex systems. Numerical simulation is used to validate the effectiveness of the numerical approximating strategy.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Iz‐iddine EL‐Fassi, Juan J. Nieto, Masakazu Onitsuka
Richards in [35] proposed a modification of the logistic model to model growth of biological populations. In this paper, we give a new representation (or characterization) of the solution to the Richards‐type fractional differential equation for , where is a continuously differentiable function on and is a positive real constant. The obtained representation of the solution can be used effectively for computational and analytic purposes. This study improves and generalizes the results obtained on fractional logistic ordinary differential equation.
{"title":"A new representation for the solution of the Richards‐type fractional differential equation","authors":"Iz‐iddine EL‐Fassi, Juan J. Nieto, Masakazu Onitsuka","doi":"10.1002/mma.10394","DOIUrl":"https://doi.org/10.1002/mma.10394","url":null,"abstract":"Richards in [35] proposed a modification of the logistic model to model growth of biological populations. In this paper, we give a new representation (or characterization) of the solution to the Richards‐type fractional differential equation for , where is a continuously differentiable function on and is a positive real constant. The obtained representation of the solution can be used effectively for computational and analytic purposes. This study improves and generalizes the results obtained on fractional logistic ordinary differential equation.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present article establishes a novel transform known as Clifford‐valued linear canonical wavelet transform which is intended to represent ‐dimensional Clifford‐valued signals at various scales, locations, and orientations. The suggested transform is capable of representing signals in the Clifford domain in addition to inheriting the characteristics of the Clifford wavelet transform. In the beginning, we demonstrate the proposed transform by the help of ‐dimensional difference of Gaussian wavelets. We then establish the fundamental properties of the proposed transform like Parseval's formula, inversion formula, and characterization of its range using Clifford linear canonical transform and its convolution. To conclude our work, we derive an analog of Heisenberg's and local uncertainty inequalities for the proposed transform.
{"title":"Clifford‐valued linear canonical wavelet transform and the corresponding uncertainty principles","authors":"Shahbaz Rafiq, Mohammad Younus Bhat","doi":"10.1002/mma.10468","DOIUrl":"https://doi.org/10.1002/mma.10468","url":null,"abstract":"The present article establishes a novel transform known as Clifford‐valued linear canonical wavelet transform which is intended to represent ‐dimensional Clifford‐valued signals at various scales, locations, and orientations. The suggested transform is capable of representing signals in the Clifford domain in addition to inheriting the characteristics of the Clifford wavelet transform. In the beginning, we demonstrate the proposed transform by the help of ‐dimensional difference of Gaussian wavelets. We then establish the fundamental properties of the proposed transform like Parseval's formula, inversion formula, and characterization of its range using Clifford linear canonical transform and its convolution. To conclude our work, we derive an analog of Heisenberg's and local uncertainty inequalities for the proposed transform.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In semiarid areas, the positive feedback effect of vegetation and soil moisture plays an indispensable role in the water absorption process of plant roots. In addition, vegetation can absorb water through the nonlocal interaction of roots. Therefore, in this article, we consider how the interactions between cross‐diffusion and nonlocal delay affect vegetation growth. Through mathematical analysis, the conditions for the occurrence of the Turing pattern in the water–vegetation model are obtained. Meanwhile, using the multi‐scale analysis method, the amplitude equation near the Turing bifurcation boundary is obtained. By analyzing the stability of the amplitude equation, the conditions for the appearance of Turing patterns such as stripes, hexagons, and mixtures of stripes and hexagons are determined. Some numerical simulations are given to illustrate the analytical results, especially the evolution processes of vegetation patterns depicted under different parameters.
{"title":"Pattern dynamics in a water–vegetation model with cross‐diffusion and nonlocal delay","authors":"Gaihui Guo, Jing You, Khalid Ahmed Abbakar","doi":"10.1002/mma.10480","DOIUrl":"https://doi.org/10.1002/mma.10480","url":null,"abstract":"In semiarid areas, the positive feedback effect of vegetation and soil moisture plays an indispensable role in the water absorption process of plant roots. In addition, vegetation can absorb water through the nonlocal interaction of roots. Therefore, in this article, we consider how the interactions between cross‐diffusion and nonlocal delay affect vegetation growth. Through mathematical analysis, the conditions for the occurrence of the Turing pattern in the water–vegetation model are obtained. Meanwhile, using the multi‐scale analysis method, the amplitude equation near the Turing bifurcation boundary is obtained. By analyzing the stability of the amplitude equation, the conditions for the appearance of Turing patterns such as stripes, hexagons, and mixtures of stripes and hexagons are determined. Some numerical simulations are given to illustrate the analytical results, especially the evolution processes of vegetation patterns depicted under different parameters.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gennaro Ciampa, Giulio G. Giusteri, Alessio G. Soggiu
We introduce models for viscoelastic materials, both solids and fluids, based on logarithmic stresses to capture the elastic contribution to the material response. The matrix logarithm allows to link the measures of strain, that naturally belong to a multiplicative group of linear transformations, to stresses, that are additive elements of a linear space of tensors. As regards the viscous stresses, we simply assume a Newtonian constitutive law, but the presence of elasticity and plastic relaxation makes the materials non‐Newtonian. Our aim is to discuss the existence of weak solutions for the corresponding systems of partial differential equations in the nonlinear large‐deformation regime. The main difficulties arise in the analysis of the transport equations necessary to describe the evolution of tensorial measures of strain. For the solid model, we only need to consider the equation for the left Cauchy–Green tensor, while for the fluid model, we add an evolution equation for the elastically‐relaxed strain. Due to the tensorial nature of the fields, available techniques cannot be applied to the analysis of such transport equations. To cope with this, we introduce the notion of charted weak solution, based on non‐standard a priori estimates, that lead to a global‐in‐time existence of solutions for the viscoelastic models in the natural functional setting associated with the energy inequality.
{"title":"Viscoelasticity, logarithmic stresses, and tensorial transport equations","authors":"Gennaro Ciampa, Giulio G. Giusteri, Alessio G. Soggiu","doi":"10.1002/mma.10469","DOIUrl":"https://doi.org/10.1002/mma.10469","url":null,"abstract":"We introduce models for viscoelastic materials, both solids and fluids, based on logarithmic stresses to capture the elastic contribution to the material response. The matrix logarithm allows to link the measures of strain, that naturally belong to a multiplicative group of linear transformations, to stresses, that are additive elements of a linear space of tensors. As regards the viscous stresses, we simply assume a Newtonian constitutive law, but the presence of elasticity and plastic relaxation makes the materials non‐Newtonian. Our aim is to discuss the existence of weak solutions for the corresponding systems of partial differential equations in the nonlinear large‐deformation regime. The main difficulties arise in the analysis of the transport equations necessary to describe the evolution of tensorial measures of strain. For the solid model, we only need to consider the equation for the left Cauchy–Green tensor, while for the fluid model, we add an evolution equation for the elastically‐relaxed strain. Due to the tensorial nature of the fields, available techniques cannot be applied to the analysis of such transport equations. To cope with this, we introduce the notion of charted weak solution, based on non‐standard a priori estimates, that lead to a global‐in‐time existence of solutions for the viscoelastic models in the natural functional setting associated with the energy inequality.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For the first time, we propose an efficient difference spectral approximation for Allen–Cahn equation in a circular domain. Firstly, we introduce the polar coordinate transformation and derive the equivalent form of Allen–Cahn equation under this coordinate system, as well as the corresponding essential polar condition. Then, by using first‐order Euler and second‐order backward difference methods in the temporal direction, we deduce the first‐order and second‐order semi‐implicit schemes, based on which the first‐order and second‐order fully discrete schemes are established by employing Legendre‐Fourier spectral approximation in the spatial direction. In addition, the energy stability and error estimations for the two types of numerical schemes are theoretically proved. Finally, we provide some numerical examples, the results of which demonstrate the stability and convergence of the algorithm.
{"title":"Stability analysis and error estimation based on difference spectral approximation for Allen–Cahn equation in a circular domain","authors":"Zhenlan Pan, Jihui Zheng, Jing An","doi":"10.1002/mma.10481","DOIUrl":"https://doi.org/10.1002/mma.10481","url":null,"abstract":"For the first time, we propose an efficient difference spectral approximation for Allen–Cahn equation in a circular domain. Firstly, we introduce the polar coordinate transformation and derive the equivalent form of Allen–Cahn equation under this coordinate system, as well as the corresponding essential polar condition. Then, by using first‐order Euler and second‐order backward difference methods in the temporal direction, we deduce the first‐order and second‐order semi‐implicit schemes, based on which the first‐order and second‐order fully discrete schemes are established by employing Legendre‐Fourier spectral approximation in the spatial direction. In addition, the energy stability and error estimations for the two types of numerical schemes are theoretically proved. Finally, we provide some numerical examples, the results of which demonstrate the stability and convergence of the algorithm.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper investigates the problems of exponential stability for a class of quaternion‐valued memristor‐based neural networks. By using M‐matrix theory and fixed point theorem, the existence and uniqueness of the equilibrium point of quaternion‐valued neural network are proved, respectively. Then, by combining M‐matrix with exponential stability theory, a non‐factorization method is obtained by using some inequality techniques to give the effective conditions of global exponential stability of quaternion‐valued memristor‐based neural network with time‐varying delay. Finally, numerical examples are given to demonstrate the validity of the derived results.
本文研究了一类基于四元数值忆阻器的神经网络的指数稳定性问题。利用 M 矩阵理论和定点定理，分别证明了四元数值神经网络平衡点的存在性和唯一性。然后，将 M 矩阵与指数稳定性理论相结合，利用一些不等式技术获得了一种非因子化方法，给出了具有时变延迟的基于四元数值忆阻器的神经网络的全局指数稳定性的有效条件。最后，给出了数值示例来证明推导结果的正确性。
{"title":"Exponential stability of a class of quaternion‐valued memristor‐based neural network with time‐varying delay via M‐matrix","authors":"Shengye Wang, Yanchao Shi, Jun Guo","doi":"10.1002/mma.10486","DOIUrl":"https://doi.org/10.1002/mma.10486","url":null,"abstract":"This paper investigates the problems of exponential stability for a class of quaternion‐valued memristor‐based neural networks. By using M‐matrix theory and fixed point theorem, the existence and uniqueness of the equilibrium point of quaternion‐valued neural network are proved, respectively. Then, by combining M‐matrix with exponential stability theory, a non‐factorization method is obtained by using some inequality techniques to give the effective conditions of global exponential stability of quaternion‐valued memristor‐based neural network with time‐varying delay. Finally, numerical examples are given to demonstrate the validity of the derived results.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we concentrate on the Bayesian inversion of a dispersion‐dominated fractional Helmholtz (DDFH) equation, which has been introduced in studies concerning seismic exploration. To establish the inversion theory, we meticulously examine the DDFH equation. We transform it into a system comprising both fractional‐ and integer‐order elliptic equations, extending the conventional definition of the spectral fractional Laplace operator to accommodate non‐homogeneous boundary conditions. Subsequently, we establish the well‐posedness theory for scenarios involving both small and large wavenumbers. Our proof hinges upon the regularity attributes of select fractional elliptic equations and capitalizes fully on the structural peculiarities of the elliptic system, which distinguish it from classical cases. Thereafter, we focus on the inverse medium scattering problem pertinent to the DDFH equation, framed within the Bayesian statistical framework. We address two scenarios: one devoid of model reduction errors and another characterized by such errors—arising from the implementation of certain absorbing boundary conditions. More precisely, based on the properties of the forward operator, well‐posedness of the posterior measures have been proved in both cases, which provide an opportunity to quantify the uncertainties of this problem.
{"title":"Bayesian inversion of a fractional elliptic system derived from seismic exploration","authors":"Yujiao Li","doi":"10.1002/mma.10474","DOIUrl":"https://doi.org/10.1002/mma.10474","url":null,"abstract":"In this paper, we concentrate on the Bayesian inversion of a dispersion‐dominated fractional Helmholtz (DDFH) equation, which has been introduced in studies concerning seismic exploration. To establish the inversion theory, we meticulously examine the DDFH equation. We transform it into a system comprising both fractional‐ and integer‐order elliptic equations, extending the conventional definition of the spectral fractional Laplace operator to accommodate non‐homogeneous boundary conditions. Subsequently, we establish the well‐posedness theory for scenarios involving both small and large wavenumbers. Our proof hinges upon the regularity attributes of select fractional elliptic equations and capitalizes fully on the structural peculiarities of the elliptic system, which distinguish it from classical cases. Thereafter, we focus on the inverse medium scattering problem pertinent to the DDFH equation, framed within the Bayesian statistical framework. We address two scenarios: one devoid of model reduction errors and another characterized by such errors—arising from the implementation of certain absorbing boundary conditions. More precisely, based on the properties of the forward operator, well‐posedness of the posterior measures have been proved in both cases, which provide an opportunity to quantify the uncertainties of this problem.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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