Analyzing stability of a delay differential equation involving two delays

Analysis of the systems involving delay is a popular topic among applied scientists. In the present work, we analyze the generalized equation $D^{\alpha} x(t) = g\left(x(t-\tau_1), x(t-\tau_2)\right)$ involving two delays viz. $\tau_1\geq 0$ and $\tau_2\geq 0$. We use the the stability conditions to propose the critical values of delays. Using examples, we show that the chaotic oscillations are observed in the unstable region only. We also propose a numerical scheme to solve such equations.Comment: 10 pages, 7 figure

A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations

In this paper we introduce a numerical method for solving nonlinear Volterra integro-differential equations. In the first step, we apply implicit trapezium rule to discretize the integral in given equation. Further, the Daftardar-Gejji and Jafari technique (DJM) is used to find the unknown term on the right side. We derive existence-uniqueness theorem for such equations by using Lipschitz condition. We further present the error, convergence, stability and bifurcation analysis of the proposed method. We solve various types of equations using this method and compare the error with other numerical methods. It is observed that our method is more efficient than other numerical methods

Analysis of solution trajectories of linear fractional order systems

The behavior of solution trajectories usually changes if we replace the classical derivative in a system by a fractional one. In this article, we throw a light on the relation between two trajectories $X(t)$ and $Y(t)$ of such a system, where the initial point $Y(0)$ is at some point $X(t_1)$ of trajectory $X(t)$. In contrast with classical systems, trajectories $X$ and $Y$ do not follow the same path. Further, we provide a Frenet apparatus of both trajectories in various cases and discuss their effect.Comment: 19 pages, 17 figure

Hybrid functions approach to solve a class of Fredholm and Volterra integro-differential equations

In this paper, we use a numerical method that involves hybrid and block-pulse functions to approximate solutions of systems of a class of Fredholm and Volterra integro-differential equations. The key point is to derive a new approximation for the derivatives of the solutions and then reduce the integro-differential equation to a system of algebraic equations that can be solved using classical methods. Some numerical examples are dedicated for showing efficiency and validity of the method that we introduce

Non-equilibrium critical behavior : An extended irreversible thermodynamics approach

Critical phenomena in non-equilibrium systems have been studied by means of a wide variety of theoretical and experimental approaches. Mode-coupling, renormalization group, complex Lie algebras and diagrammatic techniques are some of the usual theoretical tools. Experimental studies include light and inelastic neutron scattering, X-ray photon correlation spectroscopy, microwave interferometry and several other techniques. Nevertheless no conclusive reatment has been developed from the basic principles of a thermodynamic theory of irreversible processes. We have developed a formalism in which we obtain correlation functions as field averages of the associated functions. By applying such formalism we attempt to find out if the resulting correlation functions will inherit the mathematical properties (integrability, generalized homogeneity, scaling laws) of its parent potentials, and we will also use these correlation functions to study the behavior of macroscopic systems far from equilibrium, specially in the neighborhood of critical points or dynamic phase transitions. As a working example we will consider the mono-critical behavior of a non-equilibrium binary fluid mixture close to its consolute point.Comment: 23 pages, 3 figures, 1 tabl

Semen analysis and sperm function parameters in patients with infertility in Navi Mumbai and Panvel region

Background: Although semen analysis is routinely used to evaluate male partner in infertile couples, infertility and problems of impaired fecundity have been a concern through ages and is also a significant clinical problem today, which affects 8-12% of couples worldwide. Aim of the study was to study different semen parameters in male factor infertility (MFI) and thus increasing the awareness regarding same.Methods: This is cross sectional study conducted between period of September 2016 to December 2018. Semen of 150 patients were studied and results were analysed as per recent WHO (2010) criteria.Results: The present study included 150 patients whose age ranged from 24 to 51 years. Patients were divided into different age groups and sperm count was studied in each group. Abnormal sperm morphology was studied with respect to sperm head, neck, tail defects and combined defects. Sperm deformity index (SDI) and Teratozoospermic index (TZI) were calculated. Other parameters including semen volume, pH, liquefaction time, sperm vitality and motility were also studied which showed significant variations. Conclusions: Although semen analysis is first and most informative investigation for evaluation of male factor infertility, studying individual semen parameters and sperm function and increasing its awareness in general population especially in developing countries is equally important. Besides, it is necessary to acknowledge its limitation with respect to collection, processing, evaluation and biological variation of samples. Also, a normal semen analysis may not prove successful fertility potential of an individual

Osteosarcoma of jaw, common entity at an uncommon site: a rare case report

Osteosarcoma is the most common primary malignant bone tumor. Jaw is an uncommon site. The etiopathogenesis is still unknown. We present a case of 47-year-old female who underwent left partial maxillectomy as the lesion increased in size over one month after removal of a tooth. Prior to the surgery, computed tomography (CT) scan was suggestive of left maxillary sinus carcinoma and biopsy report was suggestive of spindle cell lesion. Partial maxillectomy specimen received in which size of the tumor was 5.5×5.5×4 cm with a pearly-white, solid, homogenous cut surface with gritty sensation on cutting. On histopathology it turned out to be osteosarcoma of the jaw. Mouse double minute 2 homolog (MDM2) and special AT-rich sequence-binding protein 2 (SATB2) was done to confirm the diagnosis of osteosarcoma. The patient was given chemotherapy after the confirmation and at present the patient is doing well

Singular points in the solution trajectories of fractional order dynamical systems

Dynamical systems involving non-local derivative operators are of great importance in Mathematical analysis and applications. This article deals with the dynamics of fractional order systems involving Caputo derivatives. We take a review of the solutions of linear dynamical systems ${}_0^C\mathrm{D}_t^\alpha X(t)=AX(t)$, where the coefficient matrix $A$ is in canonical form. We describe exact solutions for all the cases of canonical forms and sketch phase portraits of planar systems. We discuss the behavior of the trajectories when the eigenvalues $\lambda$ of $A$ are at the boundary of stable region i.e. $|arg(\lambda)|=\frac{\alpha\pi}{2}$. Further, we discuss the existence of singular points in the trajectories of such systems in a region of $\mathbb{C}$ viz. Region II. It is conjectured that there exists singular point in the solution trajectories if and only if $\lambda\in$ Region II.Comment: 12 pages, 22 figure