Analytical solutions of the multi-term modified power law wave equations in a finite domain

Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development. In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n) (n > 2), respectively. Analytical solutions of the multi-term modified power law wave equations are derived. These new techniques are based on Luchko’s Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi term time-space fractional models including fractional Laplacian.

Numerical methods for solving the multi-term time-fractional wave-diffusion equations

In this paper, the multi-term time-fractional wave diffusion equations are considered. The multiterm time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.

Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain

Multi-term time-fractional differential equations have been used for describing important physical phenomena. However, studies of the multi-term time-fractional partial differential equations with three kinds of nonhomogeneous boundary conditions are still limited. In this paper, a method of separating variables is used to solve the multi-term time-fractional diffusion-wave equation and the multi-term time-fractional diffusion equation in a finite domain. In the two equations, the time-fractional derivative is defined in the Caputo sense. We discuss and derive the analytical solutions of the two equations with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin conditions, respectively.

Tooth fracture risk analysis based on a new finite element dental structure models using micro-CT data

The finite element (FE) analysis is an effective method to study the strength and predict the fracture risk of endodontically-treated teeth. This paper presents a rapid method developed to generate a comprehensive tooth FE model using data retrieved from micro-computed tomography (μCT). With this method, the inhomogeneity of material properties of teeth was included into the model without dividing the tooth model into different regions. The material properties of the tooth were assumed to be related to the mineral density. The fracture risk at different tooth portions was assessed for root canal treatments. The micro-CT images of a tooth were processed by a Matlab software programme and the CT numbers were retrieved. The tooth contours were obtained with thresholding segmentation using Amira. The inner and outer surfaces of the tooth were imported into Solidworks and a three-dimensional (3D) tooth model was constructed. An assembly of the tooth model with the periodontal ligament (PDL) layer and surrounding bone was imported into ABAQUS. The material properties of the tooth were calculated from the retrieved CT numbers via ABAQUS user's subroutines. Three root canal geometries (original and two enlargements) were investigated. The proposed method in this study can generate detailed 3D finite element models of a tooth with different root canal enlargements and filling materials, and would be very useful for the assessment of the fracture risk at different tooth portions after root canal treatments.

Molecular structure of the mineral svanbergite SrAl3(PO 4,SO4)2(OH)6 - A vibrational spectroscopic study

The mineral svanbergite SrAl 3(PO 4,SO 4) 2(OH) 6 is a hydroxy phosphate-sulphate mineral belonging to the beudantite subgroup of alunites and has been characterised by vibrational spectroscopy. Bands at various wavenumbers were assigned to the different vibrational modes of svanbergite, which were then associated with the structure of the mineral. Bands were primarily assigned to phosphate and sulphate stretching and bending modes. Two symmetric stretching modes for both phosphate and sulphate supported the concept of non-equivalent phosphate and sulphate units in the mineral structure. Bands in the OH stretching region enabled hydrogen bond distances to be calculated. Comparison of the hydrogen bond distances and the calculated hydrogen bond distances from the structure models indicates that hydrogen bonding in svanbergite occurs between the two OH units rather than OH to SO42- units.

Zoledronic acid preserves bone structure and increases survival but does not limit tumour incidence in a prostate cancer bone metastasis model

Background The bisphosphonate, zoledronic acid (ZOL), can inhibit osteoclasts leading to decreased osteoclastogenesis and osteoclast activity in bone. Here, we used a mixed osteolytic/osteoblastic murine model of bone-metastatic prostate cancer, RM1(BM), to determine how inhibiting osteolysis with ZOL affects the ability of these cells to establish metastases in bone, the integrity of the tumour-bearing bones and the survival of the tumour-bearing mice. Methods The model involves intracardiac injection for arterial dissemination of the RM1(BM) cells in C57BL/6 mice. ZOL treatment was given via subcutaneous injections on days 0, 4, 8 and 12, at 20 and 100 µg/kg doses. Bone integrity was assessed by micro-computed tomography and histology with comparison to untreated mice. The osteoclast and osteoblast activity was determined by measuring serum tartrate-resistant acid phosphatase 5b (TRAP 5b) and osteocalcin, respectively. Mice were euthanased according to predetermined criteria and survival was assessed using Kaplan Meier plots. Findings Micro-CT and histological analysis showed that treatment of mice with ZOL from the day of intracardiac injection of RM1(BM) cells inhibited tumour-induced bone lysis, maintained bone volume and reduced the calcification of tumour-induced endochondral osteoid material. ZOL treatment also led to a decreased serum osteocalcin and TRAP 5b levels. Additionally, treated mice showed increased survival compared to vehicle treated controls. However, ZOL treatment did not inhibit the cells ability to metastasise to bone as the number of bone-metastases was similar in both treated and untreated mice. Conclusions ZOL treatment provided significant benefits for maintaining the integrity of tumour-bearing bones and increased the survival of tumour bearing mice, though it did not prevent establishment of bone-metastases in this model. From the mechanistic view, these observations confirm that tumour-induced bone lysis is not a requirement for establishment of these bone tumours.