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Issue 1
Volume 8 (2020)

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Authors |
: | Shahanaj Pervin and Md. Aynul Habib |

Title |
: | Solitary Wave Solutions to the Korteweg-de Vries (KdV) and the Modified Regularized Long Wave (MRLW) Equations |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 1−5 |

**Abstract: ** The main aim of this paper is to demonstrate the use of cosine function method for nonlinear partial differential equations such as Korteweg-de Vries (kdV) equation and Modified Regularized Long Wave (MRLW) equation. The cosine function method provides a mathematical powerful tool for obtaining exact travelling wave solution of these equations. The significance of obtained solutions gives credence to the explanation and understanding of related physical phenomena.

**Keywords: ** Korteweg-de Vries (kdV) equation and the Modified Regularized Long Wave (MRLW) equation, Cosine-Function method.

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Authors |
: | R. A. Mundewadi and B. A. Mundewadi |

Title |
: | Numerical Solution of Linear and Nonlinear Integral and Integro-Differential Equations using Biorthogonal Spline Wavelet Transform Method |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 7−32 |

**Abstract: ** Biorthogonal spline wavelet method is proposed for the numerical solution of Linear and nonlinear integral and integro-differential equations. Biorthogonal spline wavelet filter coefficients have the prolongation and restriction operators. The performance of the proposed method is better than the existing ones in terms of super convergence with low computational time. Some of the test problems are demonstrated for the applicability and efficiency of the scheme.

**Keywords: ** Biorthogonal spline wavelets; Filter coefficients; Multigrid Method; Full-approximation scheme; Integral equations; Integro-differential equations.

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Authors |
: | Himanshu Tiwari and Subhashish Biswas |

Title |
: | Non linear contraction mapping and its application in Dynamic Programming |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 33−39 |

**Abstract: ** We introduce the definition of $(\alpha -\beta)$-tripled fixed point in the space of the bounded functions on a set S and we present a result about the existence and uniqueness of such points. Moreover, as an application of our result, we study the problem of existence and uniqueness of solutions for a class of systems of functional equations arising in dynamic programming.

**Keywords: ** $(\alpha -\beta)$-tripled fixed point, real valued function, functional equations, dynamic programming.

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Authors |
: | S. G. Shete and R. G. Metkar |

Title |
: | Minimal and Maximal Solution for Nonlinear Volterra Type Random Integral Equations |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 41−47 |

**Abstract: ** In this research paper the existence of minimal and maximal solution for Nonlinear Volterra type random integral equations is proved under some contraction, continuity and monotonicity conditions.

**Keywords: ** Random Integral Equals, Minimal and Maximal Solutions, Volterra Integral equations.

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Authors |
: | Jaydeep Parejiya, Yesha Hathi and Patat Sarman |

Title |
: | A Note on a Line Graph of the Zero Divisor Graph of a Commutative Ring |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 49−53 |

**Abstract: ** The rings considered in this article are commutative with identity $ 1 \neq 0 $. Recall that the zero divisor graph of a ring $ R $ is a simple undirected graph whose vertex set is the set of all nonzero zero divisors of the ring $ R $ and two distinct vertices $ x, y $ are adjacent in this graph if and only if $xy = 0$. In this article we studied the line graph of the zero divisor graph of a ring and we proved some results regarding the diameter of the line graph.

**Keywords: ** Line graph, zero divisor graph, diameter of a graph.

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Authors |
: | Kanchan Barman and Subhashish Biswas |

Title |
: | Fixed Point Contractive Mapping of Wardoski Type and its Application |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 55−67 |

**Abstract: ** In the paper, we introduce a new concept of soft contraction of Wardoski type which is generalization of Banach contractive condition and prove a soft fixed point theorem which generalizes Banach contraction principle in a different ways. We also give some examples which shows
the validity of our results.

**Keywords: ** Fixed point, metric space, F-contraction.

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Authors |
: | Pravin Vadhel |

Title |
: | Some Results on Intersection Graphs of Ideals of Commutative Rings |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 69−75 |

**Abstract: ** The rings considered in this article are commutative with identity which
admit at least one nonzero proper ideal. Let $R$ be a ring. Recall that the intersection graph of ideals of $R$, denoted by $G(R)$, is an undirected simple graph whose vertex set is the set of all nontrivial ideals of $R$ (an ideal $I$ of $R$ is said to be nontrivial if $I\notin {\{(0),R}\})$ and distinct vertices $I,J$ are joined by an edge in $G(R)$ if and only if $I \cap J\,\, \ne (0)$. Let $r \in \mathbb{N}$. The aim of this article is to characterize rings $R$ such that $G(R)$ is either bipartite or 3-partite.

**Keywords: ** Bipartite graph, Quasilocal ring, Special principal ideal ring(SPIR).

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Authors |
: | E. Azuaba, J. M. Orverem, Y. M. Kura and U. Jnr. Dahiru |

Title |
: | Mathematical Approach for Modelling Malaria Disease in the Presence of Drug Therapy and Treatment |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 77−88 |

**Abstract: ** An $S-L-B-I-Q-T$ epidemic mathematical model incorporating drug therapy and treatment is investigated for malaria disease. We obtained the Disease Free Equilibrium (DFE) points and compute the basic reproduction number ($R_{0} $). The local and global stability of the Disease Free Equilibrium was analyzed using Jacobian matrix stability techniques and Lyapunov function respectively. The local and global stability was asymptotically stable if $R_{0} <1$ and $R_{0} \le 1$ respectively. Sensitivity analysis of $R_{0} $ for drug therapy and treatment showed that $R_{0} $ is strictly a decreasing function of $\sigma _{3} ,\theta ,\nu ,\tau $ and $p$. The numerical simulation of $R_{0} $ and control parameters of the model were presented graphically. The findings of this study strongly suggest a combination of effective drug therapy and treatment as a crucial strategy to control the malaria disease.

**Keywords: ** Malaria disease, Drug therapy, Treatment, Sensitivity, Stability.

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Authors |
: | M. P. Sunil and J. Suresh Kumar |

Title |
: | On Fuzzy Distance in Fuzzy Graphs |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 89−93 |

**Abstract: ** Distance and distance related concepts are well studied and used in many applications of graph theory. In this we paper, we introduce and study a new idea of fuzzy distance in fuzzy graphs and investigate the concepts like fuzzy eccentricity, fuzzy centre, etc. and study the fuzzy distance matrix and centre of fuzzy graphs.

**Keywords: ** Fuzzy graphs, Fuzzy Distance, Eccentricity, Centre, Self-centered fuzzy graph.

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Authors |
: | Ram Milan Singh |

Title |
: | A Generalized Method to Find the Square Root of Matrix Whose Characteristic Equation is Quadratic |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 95−99 |

**Abstract: ** In this paper, we generalized the method to calculating the square root of matrix whose characteristic is quadratic and how to Cayley-Hamilton theorem may be used to determine the formula for all square root of matrix whose order is $2\times2$.

**Keywords: ** Eigen values, Matrix equation, Square root of matrix.

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Authors |
: | Ram Milan Singh |

Title |
: | Generalized Method to Find the Generators of Matrix Algebras when its Dimension 2 and 3 |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 101−105 |

**Abstract: ** Let A be an algebraically closed field of characteristic zero and consider a set of $2 \times 2$ or $3 \times 3$ matrices. Using a theorem of Shemesh, we give conditions for when the matrices in the set generate the full matrix algebra.

**Keywords: ** Generator, Matrix, Common left eigen vector, Sub Algebra.

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Authors |
: | Wanjala Victor, R. K. Obogi and M. O. Okoya |

Title |
: | On N-Metric Equivalence of Operators |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 107−109 |

**Abstract: ** In this paper, we introduce a new equivalence relation, the class of n-metrically equivalent operators and examine some of the properties they get to enjoy. We also study their relation to other classes of operators like quasinormal, k-quasinormal and metrically equivalent operators.

**Keywords: ** n-metric equivalence, metric equivalence, quasinormal and k-quasinormal operators.

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Authors |
: | V. Ananthi and K. Bhuvaneswari |

Title |
: | Further Study on $\omega$-closed Sets |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 111−122 |

**Abstract: ** The aim of this paper is to prove the notion called semi-$\omega_\alpha$-open sets which is weaker than $\alpha-\omega_\alpha$-open sets and stronger than $\beta-\omega_\alpha$-open sets. Also we introduce and investigate some new generalized classes of $\tau_\omega$.

**Keywords: ** $\omega$-closed Sets, Semi-$\omega_\alpha$-open sets, $\beta-\omega_\alpha$-open sets.

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Authors |
: | Amulya Sharma |

Title |
: | Introduction to Abelian Spaces: A Connection with Topology |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 123−125 |

**Abstract: ** An abelian group is an algebraic property of an algebraic structure, but it can also be considered as a metric space. This paper establishes a new concept ``Abelian Space'', a space containing a set and an operator which is not only an abelian group, but can also be viewed as a metric space. In this paper, with reference to the symmetric difference operator between two sets, it is proved that an abelian group can also be considered as an abelian ``metric'' space.

**Keywords: ** Abelian Group, Symmetric Difference, Abelian Space.

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Authors |
: | G. Naga Malleswari, S. Sreenivasulu and G. Shobhalatha |

Title |
: | Centralizing Properties of $\left( \alpha,1\right) $ Derivations in Semiprime Rings |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 127−132 |

**Abstract: ** Let R be a semiprime ring with center Z, S be a non-empty subset of R, $\alpha$ be an endomorphism on R and $d$ be an $\left( \alpha,1\right)$ derivation of R.
A mapping $f$ from R into itself is called centralizing on S if $ \left[ f(x),x\right] \in Z $, for\ all\ $ x\in S $.
In the present paper, we study some centralizing properties of $ (\alpha,1) $ derivations in semiprime rings one of the following conditions holds:
$\left( i\right) d\left( \left[ x,y\right] \right) =\left[ x,y\right] _{\alpha,1} $, for all \ $ x,y\in R$.
$\left( ii\right) d\left( \left[ x,y\right] \right) =-\left[ x,y\right] _{\alpha,1} $, for all \ $ x,y\in R $.
$ \left( iii\right) d\left( x\right) d\left( y\right) \mp xy \in Z $, for all\ $ x,y \in R $.
$\left( iv\right) d\left( xoy\right) =\left( xoy\right) _{\alpha,1} $, for all \ $ x,y\in R $ .
$\left( v\right) d\left( xoy\right) =-\left( xoy\right) _{\alpha,1} $, for all \ $ x,y\in R $.
Also we prove that $d$ is centralizing on R if $d$ acts as a homomorphism on R and $d$ is centralizing on S if $d$ acts as an antihomomorphism on R.

**Keywords: ** Semiprime rings, $\left( \alpha,1\right)$ derivation, centralizing mappings, homomorphism and antihomomorphism.

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Authors |
: | Dr. Susanta Kumar Mohanta |

Title |
: | Virtual Tour of a Travelling Salesman |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 133−141 |

**Abstract: ** The main focus of this paper is to study the fuzzy travelling salesman problems in the true environment of fuzzy decision variables through the virtual modifier function in association with a modifier coefficient from a network of cities in a complete graph or complete digraph or connected graph.

**Keywords: ** Fuzzy decision variable; Modifier coefficient; Virtual modifier function.

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Authors |
: | Poonam Rani, Arti Sexana and Pragati Gautam |

Title |
: | Topological Structure of Quasi-Partial b-Metric Spaces |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 143−147 |

**Abstract: ** In this paper we discuss the topological properties of quasi-partial b-metric spaces. The notion of quasi-partial b-metric space was introduced and fixed point theorem and coupled fixed point theorem on this space were studied. Here the concept of quasi-partial b-metric topology is discussed and notion of product of quasi-partial b-metric spaces is also introduced.

**Keywords: ** Topological properties; b-metric spaces; fixed point theorem.

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Authors |
: | B. C. Dhage and S. S. Bellale |

Title |
: | A Local Existence Theorem for Nonlinear Perturbed Abstract Measure Integrodifferential Equations |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 149−160 |

**Abstract: ** In this paper, we prove the relevance and the local existence theorems for a class of nonlinear perturbed abstract measure integrodifferential equations via classical hybrid fixed point theorem of Dhage (2003) under weaker Lipschitz and
Carath\'eodory conditions. Our natural hypotheses and claims have also been illustrated with a numerical example.

**Keywords: ** Abstract measure integrodifferential equation; Relevance theorem; Dhage fixed point principle; Existence theorem.

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Authors |
: | Himanshu Tiwari and Subhashish Biswas |

Title |
: | An Extension of the Compression-expansion Fixed Point Theorem of Functional Type |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 161−168 |

**Abstract: ** In this article we use an interval of functional type as the underlying
set in our compression-expansion fixed point theorem argument which can
be used to exploit properties of the operator to improve conditions that
will guarantee the existence of a fixed point in applications.
An example is provided to demonstrate how intervals of functional type
can improve conditions in applications to boundary value problems.
We also show how one can use suitable $k$-contractive conditions to prove
that a fixed point in a functional-type interval is unique.

**Keywords: ** Fixed-point theorem; k-contractive; expansion; compression.

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Authors |
: | Kanchan Barman and Subhashish Biswas |

Title |
: | Meir Keeler Type Contraction in Soft Cone Ball Metric Space |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 169−186 |

**Abstract: ** In this paper, we define a new cone ball-metric and get fixed points and common fixed points for the Meir-Keeler type functions in cone ball-metric spaces.

**Keywords: ** Meir-Keeler type mapping, cone ball-metric space, common
fixed point theorem, fixed point theorem.

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Authors |
: | S. S. Khunti, J. A. Gadhiya, M. A. Chaurasiya and M. P. Rupani |

Title |
: | Antimagic Labeling of Arrow Graph, Double Arrow Graph and Globe Graph |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 187−192 |

**Abstract: ** All the Graphs consider in this article are finite, simple and undirected. In this article we investigate antimagic labeling of Arrow Graph $(A^2_n)$, Double Arrow Graph $(DA^2_n)$ and Globe graph $(Gl(n))$.

**Keywords: ** Antimagic labeling, Antimagic graph, Arrow graph, Double arrow graph.

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Authors |
: | Anurudra Y. Shete and Gitaram Pawar |

Title |
: | Existence Result for Riemann-Liouville Fractional Differential Equation with Boundary Condition |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 193−200 |

**Abstract: ** Investigation of existence property of Riemann-Liouville Fractional Differential Equation with Boundary Condition is done in this paper
${-D}^p_{0+}x\left(t\right)=f\left(t,x\left(t\right)\right)$, $0< t< 1$;
$x\left(0\right)=x'\left(0\right)=x''\left(0\right)=\dots =x^{\left(n-2\right)}\left(0\right)=0, x\left(1\right)=\lambda \int^1_0{x\left(s\right)ds}$
the technique we have employed is coupled lower and upper solutions with fixed point theory on cone, where $2\le n-1

**Keywords: ** Riemann-Liouville fractional differential equation, boundary condition, fixed point theory.

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Authors |
: | Preeti Nagar and Rajendra Pandey |

Title |
: | Using Gaussian Function to Construct the Appropriate Wavelet Function for Accurate Computation of Derivative |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 201−206 |

**Abstract: ** Maurice Hasson has constructed wavelet function using maxican hat function and richardson extrapolation technique has been used in his work. In this paper the author have constructing the appropriate wavelet function by using gaussian function. This wavelet is usfull for accurate computation of derivative. we also use Richardson Extrapolation technique in our construction. We present many vanishing moment condition in our construction when being convolved in a precise manner.

**Keywords: ** Richardson extrapolation technique, Error Estimate, Taylor Series.

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Authors |
: | Bernard Mutuku Nzimbi and Stephen Wanyonyi Luketero |

Title |
: | On Unitary Quasi-Equivalence of Operators |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 207−215 |

**Abstract: ** In this paper we investigate unitary quasi-equivalence of operators in Hilbert spaces. We characterize operators that are unitarily quasi-equivalent. We also investigate equivalence relations closely related to unitary quasi-equivalence. We give and prove conditions under which unitary quasi-equivalence coincides with other operator equivalence relations.

**Keywords: ** Unitary quasi-equivalence, near-equivalence, isometric equivalence, metric equivalence, skew-normal.

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Authors |
: | K. Sowbarnia |

Title |
: | Soft $\tau_1\tau_2\ g^*s$ Closed Sets and Their Mappings in Bi Soft Topological Spaces |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 217−230 |

**Abstract: ** In this paper we introduce and study soft $\tau_1\tau_2\ g^*s$ closed sets, soft $\tau_1\tau_2\ g^*s$ continuous mappings and soft $\tau_1\tau_2\ g^*s$ irresolute mappings %soft $\tau_1\tau_2\ g^*s$ closed and open mappings and
soft $\tau_1\tau_2\ g^*s$ homeomorphisms in bi soft topological spaces.

**Keywords: ** Soft $\tau_1\tau_2\ g^*s$ closed sets, soft $\tau_1\tau_2\ g^*s$ continuous mappings, soft $\tau_1\tau_2\ g^*s$ irresolute mappings, soft $\tau_1\tau_2\ g^*s$ homeomorphisms.

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Authors |
: | V. Mathew and S. Arumugam |

Title |
: | Universal Minimal Resolving Functions in Graphs |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 231−237 |

**Abstract: ** A vertex $x$ in a connected graph $G=(V,E)$ is said to resolve a pair $\{u,v\}$ of vertices of $G$ if the distance from $u$ to $x$ is not equal to the distance from $v$ to $x$. For the pair $\{u,v\}$ of vertices of $G$ the collection of all resolving vertices is denoted by $R\{u,v\}$ and is called the resolving neighborhood for the pair $\{u,v\}$. A real valued function $g : V \rightarrow [0,1]$ is a resolving function $(RF)$ of $G$ if $g(R\{u,v\}) \geq 1$ for all distinct pair $u,v \in V$. A resolving function $g$ is minimal ($MRF$) if any function $f:V\rightarrow [0,1]$ such that $f \leq g$ and $f(v) \neq g(v)$ for at least one $v \in V$ is nota resolving function of $G.$ A minimal resolving function $(MRF)$ is called a universal minimal resolving function $(UMRF)$ if its convex combination with every other $MRF$ is again an $MRF$. Minimal resolving functions are related to the fractional metric dimension of graphs. In this paper, we initiate a study of universal minimal resolving functions of a connected graph $G$.

**Keywords: ** Metric dimension, Fractional metric dimension, Resolving set, Resolving function, Universal minimal resolving function.

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Authors |
: | V. Mathew and Ansmol George |

Title |
: | The Minimum Resolving Energy of a Graph |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 239−246 |

**Abstract: ** A subset $W$ of vertrices in a connected graph $G=(V,E)$ is called a resolving set of $G$ if all other vertices are uniquely determined by their distances in $W.$ The metric dimension $dim(G)$ of a graph $G$ is the minimum cardinality of a resolving set of $G.$ In this paper, for a minimum resolving set $R$ of a graph $G,$ we define the minimum resolving energy $E_R(G)$ of $G.$ We study this parameter for some standard graphs. Some properties of $E_R(G)$ and bounds were also obtained.

**Keywords: ** Minimum resolving set, metric dimension, minimum resolving matrix, minimum resolving eigenvalue, minimum resolving energy of a graph.

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Authors |
: | Pramod Kumar Rawat |

Title |
: | Generalized Sixth Order Mock Theta Functions and Some Identities |

Volume, Issue, Year |
: | 8(1)(2020) |

Pages |
: | 247−257 |

**Abstract: ** We have given a new generalization of sixth order mock theta functions by introducing four independent variables. We found some identities for these generalized mock theta functions. We also have given $q$-integral representation and multibasic expansion for these functions.

**Keywords: ** q-Hypergeometric Series, Continued Fraction, q-Integrals.