This paper applies the Euler and the fourth-order Runge–Kutta methods in the analysis of fractional order dynamical systems. In order to illustrate the two techniques, the numerical algorithms are applied in the solution of several fractional attractors, namely the Lorenz, Duffing and Liu systems. The algorithms are implemented with the aid of Mathematica symbolic package. Furthermore, the Lyapunov exponent is obtained based on the Euler method and applied with the Lorenz fractional attractor.
In the present study, we dilate the differential transform scheme to develop a reliable scheme for studying analytically the mutual impact of temporal and spatial fractional derivatives in Caputo’s sense. We also provide a mathematical framework for the transformed equations of some fundamental functional forms in fractal 2-dimensional space. To demonstrate the effectiveness of our proposed scheme, we first provide an elegant scheme to estimate the (mixed-higher) Caputo-fractional derivatives, and then we give an analytical treatment for several (non)linear physical case studies in fractal 2-dimensional space. The study concluded that the proposed scheme is very efficacious and convenient in extracting solutions for wide physical applications endowed with two different memory parameters as well as in approximating fractional derivatives.
A portable pneumatic video-tactile sensor for determining the local stiffness of soft tissue and the methodology for its application are considered. The expected range of local elastic modulus that can be estimated by the sensor is 100 kPa–1 MPa. The current version of the device is designed to determine the characteristics of tissues that are close in mechanical properties to the skin with subcutis and muscles. A numerical simulation of the contact between the sensor head and the soft tissue was performed using the finite-element method. Both 2D and 3D models were developed. Results of experiments with device prototype are used for approval of adequacy of mathematical modelling in case of large deformations. Simulation results can be used to create soft tissue databases, which will be required to determine the local stiffness of soft tissues by the sensor. 2D model proved to be more efficient for the chosen range of values of local stiffness of soft tissues.
This paper considers a Hantavirus infection model consisting of a system of fractional-order ordinary differential equations with logistic growth. The fractional-order model describes the spread of Hantavirus infection in a system consisting of a population of susceptible and infected mice. The existence, uniqueness, non-negativity and boundedness of the solutions are established. In addition, the local and global asymptotic stability of the equilibrium points of the fractional order system and the basic reproduction number are studied. The impact of basic reproduction number and carrying capacity on the stability of the fractional order system are also theoretically and numerically investigated.
We consider the existence, uniqueness and Ulam–Hyers–Rassias stability of solutions to the initial value problem with noninstantaneous impulses on ordered Banach spaces. The existence and uniqueness of solutions for nonlinear ordinary differential equation with noninstantaneous impulses is obtained by using perturbation technique, monotone iterative method and a new estimation technique of the measure of noncompactness under the situation that the corresponding noninstantaneous impulsive functions gi are compact and not compact, respectively. Furthermore, the UHR stability of solutions is also obtained, which provides an approach to find approximate solution to noninstantaneous impulsive equations in the sense of UHR stability.
In this paper, we constructed the variational principles for Bogoyavlensky–Konopelchenko equation, the generalized (3+1)-dimensional nonlinear wave in liquid containing gas bubbles and a new coupled Kadomtsev–Petviashvili (KP) equation via He’s semi-inverse method. Based on this formulation, we obtained the solitary wave solutions via Ritz method. We explained the properties of the soliton waves numerically by some figures. Finally, the physical interpretation for these solutions are obtained.
Mass or heat transfer may cause volume variation, and the food hydration model is one of them that undergoes hydration (or drying) conveying volume change. In this paper, the numerical approximate solution based on an integral method has been presented for soybean hydration model. Trace of the moving boundary and unknown moisture content at the center of the grain have been determined. The obtained results are well matched with numerical solutions in the literature.
In this paper, we concern with the existence of mild solution to nonlocal initial value problem for nonlinear Sobolev-type impulsive evolution equations with Hilfer fractional derivative which generalized the Riemann–Liouville fractional derivative. At first, we establish an equivalent integral equation for our main problem. Second, by means of the properties of Hilfer fractional calculus, combining measure of noncompactness with the fixed-point methods, we obtain the existence results of mild solutions with two new characteristic solution operators. The results we obtained are new and more general to known results. At last, an example is provided to illustrate the results.
We study a fractional integro-differential equation subject to multi-point boundary conditions:
where . By utilizing a new fixed point theorem of increasing concave operators defined on special sets in ordered spaces, we demonstrate existence and uniqueness of solutions for this problem. Besides, it is shown that an iterative sequence can be constructed to approximate the unique solution. Finally, the main result is illustrated with the aid of an example.