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We propose a new method to solve the initial-boundary value problem for hyperbolic-dissipative partial differential equations (PDEs) based on the spirit of LY algorithm [T.-P. Liu and S.-H. Yu, Dirichlet–Neumann kernel for hyperbolic-dissipative system in half-space, Bull. Inst. Math. Acad. Sin. 7 (2012) 477–543]. The new method can handle more general domains than that of LYs'. We convert the evolutionary PDEs into the elliptic PDEs by the Laplace transformation. Using the Laplace transformation of the fundamental solutions of the evolutionary PDEs and the image method, we can construct Green's functions for the corresponding elliptic PDEs. Finally, we obtain Green's functions for the evolutionary PDEs by inverting the Laplace transformation. As a consequence, we establish Green's functions for some basic PDEs such as the heat equation, the wave equation and the damped wave equation, in a half space and a quarter plane with various boundary conditions. On the other hand, the structure of hyperbolic-dissipative PDEs means its fundamental solution is non-symestric and hence the image method does not work. We utilize the idea of Laplace wave train introduced by Liu and Yu in [Navier–Stokes equations in gas dynamics: Greens function, singularity and well-posedness, Comm. Pure Appl. Math. 75(2) (2022) 223–348] to generalize the image method. Combining this with the notions of Rayleigh surface wave operators introduced in [S. J. Deng, W. K. Wang and S.-H. Yu, Green's functions of wave equations in ℝ + n × ℝ + , Arch. Ration. Mech. Anal. 216 (2015) 881–903], we are able to obtain the complete representations of Green's functions for the convection-diffusion equation and the drifted wave equation in a half space with various boundary conditions. [ABSTRACT FROM AUTHOR]

Geographic ranges of communities of species evolve in response to environmental, ecological, and evolutionary forces. Understanding the effects of these forces on species’ range dynamics is a major goal of spatial ecology. Previous mathematical models have jointly captured the dynamic changes in species’ population distributions and the selective evolution of fitness-related phenotypic traits in the presence of an environmental gradient. These models inevitably include some unrealistic assumptions, and biologically reasonable ranges of values for their parameters are not easy to specify. As a result, simulations of the seminal models of this type can lead to markedly different conclusions about the behavior of such populations, including the possibility of maladaptation setting stable range boundaries. Here, we harmonize such results by developing and simulating a continuum model of range evolution in a community of species that interact competitively while diffusing over an environmental gradient. Our model extends existing models by incorporating both competition and freely changing intraspecific trait variance. Simulations of this model predict a spatial profile of species’ trait variance that is consistent with experimental measurements available in the literature. Moreover, they reaffirm interspecific competition as an effective factor in limiting species’ ranges, even when trait variance is not artificially constrained. These theoretical results can inform the design of, as yet rare, empirical studies to clarify the evolutionary causes of range stabilization. [ABSTRACT FROM AUTHOR]

Recently, various evolutionary partial differential equations (PDEs) with a mixed derivative have been emerged and drawn much attention. Nonetheless, their PDE-theoretical and numerical studies are still in their early stage. In this paper, we aim at the unified framework of numerical methods for such PDEs. However, due to the presence of the mixed derivative, we cannot discuss numerical methods without some appropriate reformulation, which is mathematically challenging itself. Therefore, we first propose a novel procedure for the reformulation of target PDEs into a standard form of evolutionary equations. This contribution may become an important basis not only of numerical analysis, but also of PDE-theory. In order to illustrate this point, we establish the global well-posedness of the sine-Gordon equation. After that, we classify and discuss the spatial discretizations based on the proposed reformulation technique. As a result, we show the average-difference method is suitable for the discretization of the mixed derivative. [ABSTRACT FROM AUTHOR]

In this work, we propose a preconditioned minimal residual (MINRES) method for a class of non-Hermitian block Toeplitz systems. Namely, considering an m n -by- m n non-Hermitian block Toeplitz matrix T (n , m) with m -by- m commuting Hermitian blocks, we first premultiply it by a simple permutation matrix to obtain a Hermitian matrix and then construct a Hermitian positive definite block circulant preconditioner for the modified matrix. Under certain conditions, we show that the eigenvalues of the preconditioned matrix are clustered around ± 1 when n is sufficiently large. Due to the Hermitian nature of the modified matrix, MINRES with our proposed preconditioner can achieve theoretically guaranteed superlinear convergence under suitable conditions. In addition, we provide several useful properties of block circulant matrices with commuting Hermitian blocks, including diagonalizability and symmetrization. A generalization of our result to the multilevel block case is also provided. We in particular indicate that our work can be applied to the all-at-once systems arising from solving evolutionary partial differential equations. Numerical examples are given to illustrate the effectiveness of our preconditioning strategy. [ABSTRACT FROM AUTHOR]