On Solutions of Fractional Telegraph Model with Mittag-Leffler Kernel

In this paper, we research the fractional telegraph equation with the Atangana-Baleanu-Caputo derivative. We use the Laplace method to find the exact solution of the problems. We construct the difference schemes for the implicit finite method. We prove the stability of difference schemes for the problems by the matrix method. We demonstrate the accuracy of the method by some numerical experiments. The obtained results confirm the accuracy and effectiveness of the proposed method. Additionally, the numerical results demonstrate that the expected physical properties of the model are also observed.

Analytical approach for the temperature distribution in the casting-mould heterogeneous system

Purpose The main purpose of this paper is to calculate the analytical solution or a closed-form solution for the temperature distribution in the heterogeneous casting-mould system. Design/methodology/approach First, the authors formulate and analyze the mathematical formulation of heat conduction equation in the heterogeneous casting-mould system, with an arbitrary assumption of the ideal contact at the cast-mould contact point. Then, He-Laplace method, based on variational iteration method (VIM), Laplace transform and homotopy perturbation method (HPM), is used to elaborate the analytical solution of this system. The main focus of He-Laplace method is to find the Lagrange multiplier with an easy approach which enables the implementation of HPM very smoothly and provides the series solution very close to the exact solution. Findings An example is considered to show that He-Laplace method provides the efficient results for calculating the temperature distribution in the casting-mould heterogeneous system. Graphical representation and error distribution represents that He-Laplace method is very simple to implement and effective for casting-mould heterogeneous system. Originality/value The work in this paper is original and advanced. Specially, calculation of Lagrange multiplier for casting-mould system has not been reported in the literature for this work.

Chio’s-like Method for Calculating the Rectangular (non-square) Determinants: Computer Algorithm Interpretation and Comparison

In this paper, we present an approach for the calculation of rectangular determinants, where in addition to the mathematical formula, we also provide a computer algorithm for their calculation. Firstly, we present a method similar to Sarrus method for calculating the rectangular determinant of the order 2 × 3. Secondly, we present an approach for calculating the rectangular determinants of order m ×n by adding a row with all elements equal to one (1) in any row, as well as an application of Chio’s rule for calculating the rectangular determinants. Thirdly, we find the time complexity and comparison of the computer execution time of calculation of the rectangular determinant based on the presented algorithms and comparing them with the algorithm based on the Laplace method.

On the approximate solution of fractional-order Whitham–Broer–Kaup equations

In this paper, the Homotopy perturbation Laplace method is implemented to investigate the solution of fractional-order Whitham–Broer–Kaup equations. The derivative of fractional-order is described in Caputo’s sense. To show the reliability of the suggested method, the solution of certain illustrative examples are presented. The results of the suggested method are shown and explained with the help of its graphical representation. The solutions of fractional-order problems as well as integer-order problems are determined by using the present technique. It has been observed that the obtained solutions are in significant agreement with the actual solutions to the targeted problems. Computationally, it has been analyzed that the solutions at different fractional-orders have a higher rate of convergence to the solution at integer-order of the derivative. Due to the analytical analysis of the problems, this study can further modify the solution of other fractional-order problems.

General Response Formula for CFD Pseudo-Fractional 2D Continuous Linear Systems Described by the Roesser Model

In many engineering problems associated with various physical phenomena, there occurs a necessity of analysis of signals that are described by multidimensional functions of more than one variable such as time t or space coordinates x, y, z. Therefore, in such cases, we should consider dynamical models of two or more dimensions. In this paper, a new two-dimensional (2D) model described by the Roesser type of state-space equations will be considered. In the introduced model, partial differential operators described by the Conformable Fractional Derivative (CFD) definition with respect to the first (horizontal) and second (vertical) variables will be applied. For the model under consideration, the general response formula is derived using the inverse fractional Laplace method. Next, the properties of the solution will be considered. Usefulness of the general response formula will be discussed and illustrated by a numerical example.

Numerical calculation for coupling vibration system by Piecewise-Laplace method

To improve the reliability and accuracy of dynamic machine in design process, high precision and efficiency of numerical computation is essential means to identify dynamic characteristics of mechanical system. In this paper, a new computation approach is introduced to improve accuracy and efficiency of computation for coupling vibrating system. The proposed method is a combination of piecewise constant method and Laplace transformation, which is simply called as Piecewise-Laplace method. In the solving process of the proposed method, the dynamic system is first sliced by a series of continuous segments to reserve physical attribute of the original system; Laplace transformation is employed to separate coupling variables in segment system, and solutions of system in complex domain can be determined; then, considering reverse Laplace transformation and residues theorem, solution in time domain can be obtained; finally, semi-analytical solution of system is given based on continuity condition. Through comparison of numerical computation, it can be found that precision and efficiency of numerical results with the Piecewise-Laplace method is better than Runge-Kutta method within same time step. If a high-accuracy solution is required, the Piecewise-Laplace method is more suitable than Runge-Kutta method.

Analysis of fractional duffing oscillator

In this contribution, a simple analytical method (which is an elegant combination of a well known methods; perturbation method and Laplace method) for solving non-linear and non-homogeneous fractional differential equations is pro- posed. In particular, the proposed method was used to analysed the fractional Duffing oscillator.The technique employed in this method can be used to analyse other nonlinear fractional differential equations, and can also be extended to non- linear partial fractional differential equations.The performance of this method is reliable, effective and gives more general solution.

Spatially Resolved Estimation of Metabolic Oxygen Consumption From Optical Measurements in Cortex

The cerebral metabolic rate of oxygen (CMRO2) is an important indicator of brain function and pathology. Knowledge about its magnitude is also required for proper interpretation of the blood oxygenation level dependent (BOLD) signal measured with functional MRI (fMRI). The ability to measure CMRO2 with high spatial and temporal accuracy is thus highly desired. Traditionally the estimation of CMRO2 has been pursued with somewhat indirect approaches combining several different types of measurements with mathematical modeling of the underlying physiological processes. Given the numerous assumptions involved, questions have thus been raised about the accuracy of the resulting CMRO2 estimates. The recent ability to measure the level of oxygen (pO2) in cortex with high spatial resolution in in vivo conditions has provided a more direct way for estimating CMRO2. CMRO2 and pO2 are related via the Poisson partial differential equation. Assuming a constant CMRO2 and cylindrical symmetry around the blood vessel providing the oxygen, the so-called Krogh-Erlang formula relating the spatial pO2 profile to a constant CMRO2 value can be derived. This Krogh-Erlang formula has previously been used to estimate the average CMRO2 close to cortical blood vessels based on pO2 measurements in rats.Here we introduce a new method, the Laplace method, to provide spatial maps of CMRO2 based on the same measured pO2 profiles. The method has two key steps: First the measured pO2 profiles are spatially smoothed to reduce effects of spatial noise in the measurements. Next, the Laplace operator (a double spatial derivative) in two spatial dimensions is applied on the smoothed pO2 profiles to obtain spatially resolved CMRO2 estimates. The smoothing introduces a bias, and a balance must be found where the effects of the noise are sufficiently reduced without introducing too much bias. In this model-based study we explore this balance in situations where the ground truth, that is, spatial profile of CMRO2 is preset and thus known, and the corresponding pO2 profiles are found by solving the Poisson equation, either numerically or by taking advantage of the Krogh-Erlang formula. MATLAB code for using the Laplace method is provided.