One Solution for Many Linear Partial Differential Equations With Terms of Equal Orders

We disclose a simple and straightforward method of solving single-order linear partial differential equations. The advantage of the method is that it is applicable to any orders and the big disadvantage is that it is restricted to a single order at a time. As it is very easy compared to classical methods, it has didactic value.

An Existence Result for Henstock-Kurzweil-Stieltjes-⋄-Double Integral of Interval-Valued Functions on Time Scales

We employ the concept of interval-valued functions to state and prove an existence result for the Henstock-Kurzweil-Stieltjes-⋄-double integral on time scales.

On Some Bounds for the Exponential Integral Function

In 1934, Hopf established an elegant inequality bounding the exponential integral function. In 1959, Gautschi established an improvement of Hopf’s results. In 1969, Luke also established two inequalities with each improving Hopf’s results. In 1997, Alzer also established another improvement of Hopf’s results. In this paper, we provide two new proofs of Luke’s first inequality and as an application of this inequality, we provide a new proof and a generalization of Gautschi’s results. Furthermore, we establish some inequalities which are analogous to Luke’s second inequality and Alzer’s inequality. The techniques adopted in proving our results are simple and straightforward.

Fixed Point Theorems on Some New Types of Cyclic Contractions and Their Generalization

There are different types of contraction in the existing literature for the generalization of Banach’s contraction principle. Our aim in this paper is to generalize cyclic contraction so that it can explain all types of cyclic contraction as a particular case. Besides all contractions in the existing literature we introduce some new types of cyclic contraction before defining the generalized cyclic contraction.

Lexicographically Maximum Flows under an Arc Interdiction

Network interdiction problem arises when an unwanted agent attacks the network system to deteriorate its transshipment efficiency. Literature is flourished with models and solution approaches for the problem. This paper considers a single commodity lexicographic maximum flow problem on a directed network with capacitated vertices to study two network flow problems under an arc interdiction. In the first, the objective is to find an arc on input network to be destroyed so that the residual lexicographically maximum flow is lexicographically minimum. The second problem aims to find a flow pattern resulting lexicographically maximum flow on the input network so that the total residual flow, if an arc is destroyed, is maximum. The paper proposes strongly polynomial time solution procedures for these problems.

Time Dependent Mathematical Model of Thermoregulation in Human Dermal Parts During Sarcopenia

Sarcopenia is an illness characterized by the loss of skeletal muscle mass, and its strength occurs in aging after 50 years. Muscle mass plays a vital role in body weight and metabolism. The loses in body weight impact reducing the basal metabolic rate (BMR). The BMR affects the human body temperature due to lower metabolic heat production during sarcopenia. The present study deals with time dependent temperature variation in human dermal parts during sarcopenia. The finite element method is used to solve a one-dimensional bioheat equation. In this model, the thickness of the epidermis, dermis layers, and the BMR of different aging, are estimated. The results show the nodal temperature of the epidermis and dermis layers increases due to reducing the thickness. Further, the subcutaneous nodal temperature slightly decreases due to the cause of BMR.

Temperature Variation in Breast Tissue Model With and Without Tumor Based on Porous Media

The human body is made by 200 different types of cells, which are separated by voids. Blood supplies the nutrients and minerals to all cells within the tissue through these voids. The breast tissue is treated as a porous media in the study. Tumor includes the vascular (blood) and the extra-vascular (solid) regions. The porosity of a tumor is higher than normal tissue. The present work deals with the temperature variation of normal and tumorous breast tissue based on porous media. The finite element method is used to solve the two-dimensional bio-heat equation. The results show that the temperature profile of normal breast tissue in the porous media model is almost identical with the conventional bio-heat model at correction factor is equal to 0:6. The temperature of tumor region in the porous media model is slightly lower than the conventional bio-heat model. When the porosity is increased, the temperature of normal breast tissue is increased. But in tumorous breast tissue, the temperature is slightly increased in skin surface to anterior part of the tumor and slightly decreased in tumor region. The temperature of normal and tumorous breast tissue is increased when metabolism, blood velocity, and room temperature are increased in the porous media model. The central temperature of the tumor region reaches a steady state faster than anterior and posterior temperature of both normal and tumorous breast tissue in conventional bio-heat model and porous media model.

Adaptive Finite Element Method for Solving Poisson Partial Differential Equation

Poisson equation is an elliptic partial differential equation, a generalization of Laplace equation. Finite element method is a widely used method for numerically solving partial differential equations. Adaptive finite element method distributes more mesh nodes around the area where singularity of the solution happens. In this paper, Poisson equation is solved using finite element method in a rectangular domain with Dirichlet and Neumann boundary conditions. Posteriori error analysis is used to mark the refinement area of the mesh. Dorfler adaptive algorithm is used to refine the marked mesh. The obtained results are compared with exact solutions and displayed graphically.

Three Classical Methods to find the Cube Roots: A Connective Perspective on Lilavati, Vedic and Pande's Procedures

South Asian region has made a glorious history of mathematics. This area is considered as fer- tile land for the birth of pioneer mathematicians who developed various mathematical ideas and creations. Among them, three innovative personalities are Bhaskaracarya, Gopal Pande and Bharati Krishna Tirthaji and their specific methods to find cube root are mainly focused on this study. The article is trying to explore the comparative study among the procedures they adopt. Gopal Pande disagrees with the Bhaskaracarya's verse. He used the unitary method against that method mentioned in Bhaskaracarya's famous book Lilavati to prove his procedures. However, the Vedic method by Tirthaji was not influenced by the other two except for minor cases. In the case of practicality and simplicity, the Vedic method is more practical and simpler to understand for all mathematical learners and teachers in comparison to the other two methods.

Analytical Solution for Advection-Dispersion Equation of the Pollutant Concentration using Laplace Transformation

We present simple analytical solution for the unsteady advection-dispersion equation describing the pollutant concentration C(x; t) in one dimension. In this model the water velocity in the x-direction is taken as a linear function of x and dispersion coefficient D as zero. In this paper by taking k = 0, k is the half saturated oxygen demand concentration for pollutant decay, we can apply the Laplace transformation and obtain the solution. The variation of C(x; t) with different times t upto t → ∞ (the steady state case) is taken into account advection-dispersion equation in our study.