R.H. De Staelen

On a class of non-linear delay distributed order fractional diffusion equations

In this paper, we consider a numerical scheme for a class of non-linear time delay fractional diffusion equations with distributed order in time. This study covers the unique solvability, convergence and stability of the resulted numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme with convergence order O ( ź + ( Δ α ) 4 + h 4 ) in L ∞ -norm is the main purpose of this study. Numerical experiments are carried out to support the obtained theoretical results.

Error analysis in the reconstruction of a convolution kernel in a semilinear parabolic problem with integral overdetermination

A semilinear parabolic problem of second order with an unknown solely time-dependent convolution kernel is considered. An additional given global measurement (a space integral of the solution) ensures the existence of a unique weak solution. The unknown kernel function can be approximated by a time-discrete numerical scheme based on Backward Eulers method (Rothes method). In this contribution, an error analysis for the time discretization is performed of the existing numerical algorithm. Numerical experiments support the theoretically obtained results.

A volume averaging and overlapping domain decomposition technique to model mass transfer in textiles

A new three scale approach for textile models is suggested: a one-dimensional fiber model and a fabric model, with a meso-level in between, i.e. the yarn scale, Goessens?et?al.?(2012). For loose textile substrates this seems appropriate as the yarn level plays an important role. This is because the saturation vapor pressure will influence the release rate from the fibers, and its value will vary over the yarn cross section. Therefore, in this work we present two upscaling techniques for the three step multiscale model. The active component is tracked in the fiber, the yarn, and finally at the fabric level. At the fiber level a one-dimensional reduction to a non-linear diffusion equation is performed, and solved on an as needed basis. The outcome is upscaled via the volume averaging method and used as an input for the yarn level. At this level a one-dimensional model can be applied to calculate the concentration, which on its turn is upscaled using the overlapping domain decomposition as an input for the fabric level model.

Identification of a memory kernel in a semilinear integrodifferential parabolic problem with applications in heat conduction with memory

In this contribution, the reconstruction of a solely time-dependent convolution kernel is studied in an inverse problem arising in the theory of heat conduction for materials with memory. The missing kernel is recovered from a measurement of the average of temperature. The existence, uniqueness and regularity of a weak solution is addressed. More specific, a new numerical algorithm based on Rothes method is designed. The convergence of iterates to the exact solution is shown.

A pseudo energy-invariant method for relativistic wave equations with Riesz space-fractional derivatives

Abstract In this work, we investigate a general nonlinear wave equation with Riesz space-fractional derivatives that generalizes various classical hyperbolic models, including the sine-Gordon and the Klein–Gordon equations from relativistic quantum mechanics. A finite-difference discretization of the model is provided using fractional centered differences. The method is a technique that is capable of preserving an energy-like quantity at each iteration. Some computational comparisons against solutions available in the literature are performed in order to assess the capability of the method to preserve the invariant. Our experiments confirm that the technique yields good approximations to the solutions considered. As an application of our scheme, we provide simulations that confirm, for the first time in the literature, the presence of the phenomenon of nonlinear supratransmission in Riesz space-fractional Klein–Gordon equations driven by a harmonic perturbation at the boundary.

A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations

In this work, we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the Riesz type. The model under consideration generalizes various nonlinear wave equations, including the sine-Gordon and the nonlinear Klein–Gordon models. The system considered in this work is conservative when homogeneous Dirichlet boundary conditions are imposed. Motivated by this fact, we propose a finite-difference method based on fractional centered differences that is capable of preserving the discrete energy of the system. The method under consideration is a nonlinear implicit scheme which has various numerical properties. Among the most interesting numerical features, we show that the methodology is consistent of second order in time and fourth order in space. Moreover, we show that the technique is stable and convergent. Some numerical simulations show that the method is capable of preserving the energy of the discrete system. This characteristic of the technique is in obvious agreement with the properties of its continuous counterpart.

Bayesian inference in the uncertain EEG problem including local information and a sensor correlation matrix

We present a framework based on Bayesian inference to combine expert judgment and the problem of an uncertain conductivity in the electroencephalography (EEG) inverse problem. A three layer spherical head model with different and random layer conductivities is considered. The randomness is modeled by Legendre Polynomial Chaos. Using this Polynomial Chaos we build on previous work to obtain a correlation matrix for the error used in the likelihood function of the Bayesian procedure. We compare with a classical isotropic correlation.

A semi-linear delayed diffusion-wave system with distributed order in time

A numerical scheme for a class of non-linear distributed order fractional diffusion-wave equations with fixed time-delay is considered. The focus lies on the derivation of a linearized compact difference scheme as well as on quantitatively analyzing it. We prove unique solvability, convergence, and stability of the resulted numerical solution in L∞

Sensitivity analysis and variance reduction in a stochastic non-destructive testing problem

L_{\infty }

Reconstruction of a convolution kernel in a semilinear parabolic problem based on a global measurement

-norm by means of the discrete energy method. Numerical examples are introduced to illustrate the accuracy and efficiency of the proposed method.