Weak Convergence Rates for Spatial Spectral Galerkin Approximations of Semilinear Stochastic Wave Equations with Multiplicative Noise

Stochastic wave equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic wave equations can typically not be solved explicitly, but the literature contains a number of results which show that numerical approximation processes converge with suitable rates of convergence to solutions of such equations. In the case of approximation results for strong convergence rates, semilinear stochastic wave equations with both additive or multiplicative noise have been considered in the literature. In contrast, the existing approximation results for weak convergence rates assume that the diffusion coefficient of the considered semilinear stochastic wave equation is constant, that is, it is assumed that the considered wave equation is driven by additive noise, and no approximation results for multiplicative noise are known. The purpose of this work is to close this gap and to establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise. In particular, our weak convergence result establishes as a special case essentially sharp weak convergence rates for the continuous version of the hyperbolic Anderson model. Our method of proof makes use of the Kolmogorov equation and the Hölder-inequality for Schatten norms.

Global existence and asymptotic behavior of solutions to fractional ( p , q )-Laplacian equations

The aim of this paper is to consider the following fractional parabolic problem u t + ( − Δ ) p α u + ( − Δ ) q β u = f ( x , u ) ( x , t ) ∈ Ω × ( 0 , ∞ ) , u = 0 ( x , t ) ∈ ( R N ∖ Ω ) × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) x ∈ Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary, ( − Δ ) p α is the fractional p-Laplacian with 0 < α < 1 < p < ∞, ( − Δ ) q β is the fractional q-Laplacian with 0 < β < α < 1 < q < p < ∞, r > 1 and λ > 0. The global existence of nonnegative solutions is obtained by combining the Galerkin approximations with the potential well theory. Then, by virtue of a differential inequality technique, we give a decay estimate of solutions.

Galerkin Approximations for the Solution of Fredholm Volterra Integral Equation of Second Kind

In this research, we have introduced Galerkin method for finding approximate solutions of Fredholm Volterra Integral Equation (FVIE) of 2nd kind, and this method shows the result in respect of the linear combinations of basis polynomials. Here, BF (product of Bernstein and Fibonacci polynomials), CH (product of Chebyshev and Hermite polynomials), CL (product of Chebyshev and Laguerre polynomials), FL (product of Fibonacci and Laguerre polynomials) and LLE (product of Legendre and Laguerre polynomials) polynomials are established and considered as basis function in Galerkin method. Also, we have tried to observe the behavior of all these approximate solutions finding from Galerkin method for different problems and then a comparison is shown using some standard error estimations. In addition, we observe the error graphs of numerical solutions in Galerkin method for different problems of FVIE of second kind. GANITJ. Bangladesh Math. Soc.41.1 (2021) 1–14

Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility

We consider Galerkin approximations of eigenvalue problems for holomorphic Fredholm operator functions for which the operators do not have the structure “coercive+compact”. In this case the regularity (in the vocabulary of discrete approximation schemes) of Galerkin approximations is not unconditionally satisfied and the question of convergence is delicate. We report a technique to prove regularity of approximations which is applicable to a wide range of eigenvalue problems. The technique is based on the knowledge of a suitable Test function operator. In particular, we introduce the concepts of weak T-coercivity and T-compatibility and prove that for weakly T-coercive operators, T-compatibility of Galerkin approximations implies their regularity. Our framework can be successfully applied to analyze e.g. complex scaling/perfectly matched layer methods, problems involving sign-changing coefficients due to meta-materials and also (boundary element) approximations of Maxwell-type equations. We demonstrate the application of our framework to the Maxwell eigenvalue problem for a conductive material.

A NUMERICAL STUDY OF A HOMOGENEOUS BEAM WITH A TIP MASS

In this paper, we prove the existence and uniqueness of the weak solution of a flexible beam that is clamped at one end and free at the other; a mass is also attached to the free end of the beam. Also, we construct a finite element method, based on piecewise cubic Hermitian shape functions. Next, we derive error estimates for the semi-discrete Galerkin approximations. The results are derived from \cite{BS}. Finally, we implement the results of numerical schemes developed.

Global existence and dynamic structure of solutions for damped wave equation involving the fractional Laplacian

We consider strong damped wave equation involving the fractional Laplacian with nonlinear source. The results of global solution under necessary conditions on the critical exponent are established. The existence is proved by using the Galerkin approximations combined with the potential well theory. Moreover, we showed new decay estimates of global solution.