Communications in Nonlinear Analysis
https://www.cna-journal.com/
Communications in Nonlinear Analysisendaily1Sat, 01 Jul 2023 00:00:00 +0430Sat, 01 Jul 2023 00:00:00 +0430Investigating a Coupled System with Riemann-Liouville Fractional Derivative by Modified Monotone Iterative Technique
https://www.cna-journal.com/article_102672.html
In recent time, the area of arbitrary order differential equations (AODEs)has been considered very well. Different aspects have been investigated for the saidarea. One of the important and most warm area is devoted to study multiplicityresults along with existence and uniqueness of solutions for the said equations. In thisregard various techniques have been utilized to investigate the said area. Monotoneiterative technique (MIT) coupled with the method of extremal solutions has beenused recently to investigate multiplicity of solutions to some AODEs. In this researchwork, we deal a coupled system of nonlinear AODEs under boundary conditions (BCs)involving Riemann-Liouville fractional derivative by using fixed point theorems dueto Perov&rsquo;s and Schuader&rsquo;s to study existence and uniqueness results. Using Perove&rsquo;sfixed point theorem ensures uniqueness of solution to systems of equations, whileexistence of at least one solution is achieved by Schauder&rsquo;s fixed point theorem.Then we come across the multiplicity of solutions and establish some criteria for theiterative solutions via using updated type MIT together with the method of upper andlower solutions for the considered system of AODEs. Corresponding to multiplicityresults of solutions, we first establish two sequences of extremal solutions. One of thesequence is monotonically decreasing and converging to lower solution. On the otherhand, the other sequence is monotonically increasing and converging to the uppersolution. In last we give suitable examples to illustrate the main results.On the Number of Zeros of A Polynomial
https://www.cna-journal.com/article_174086.html
In this paper, we consider the problem of finding the maximum number of zeros of a polynomial in a prescribed region. Our theorems includes several known results in this direction as special cases.Analytical Solutions of a Class of Generalised Lane-Emden Equations: Power Series Method Versus Adomian Decomposition Method
https://www.cna-journal.com/article_174087.html
In this paper, we obtain highly accurate analytical solutions of a class of strongly nonlinear Lane-Emden equations using a power series method and the Adomian decomposition method. The nonlinear term of the proposed problem involves the integer powers of a continuous real-valued function $\Lambda(y(x))$. In each of the proposed methods, a unified result is presented for the function $\Lambda(y(x))$. The particular cases of the trigonometric functions $\Lambda(y(x))=\tan y(x)$, $\sec y(x)$ and the hyperbolic functions $\Lambda(y(x))=\tanh y(x)$, $\sech y(x)$ are considered explicitly using the proposed methods. Lane-Emden equations involving the first integer powers of these trigonometric and hyperbolic functions are given as examples to illustrate the reliability, efficiency and accuracy of the proposed methods. Numerical comparisons of the results obtained show excellent agreements between the two methods, an indication that both methods are accurate, effective, reliable and convenient in solving singular strongly nonlinear ordinary differential equations with appropriate initial conditions.Analytical Solution of the Mathematical Model of Dengue Fever by the Laplace-Adomian Decomposition Method
https://www.cna-journal.com/article_174090.html
This paper presents and rigorously analyzes a deterministic mathematical model of the Dengue virus in apopulation, incorporating a nonlinear incidence function. The model considers five compartments: susceptible (S h), symptomatic infection (Ih), asymptomatic infection (IhA), recovered (Rh), and partial immunity(S hk). The female mosquito population is divided into two compartments: susceptible (S &nu;)and infected (I&nu;).An algorithm is provided to calculate a series-type solution to the problem using the Laplace Adomian Decomposition technique. The convergence of this technique is also analyzed. Approximations of the solutionsfor various compartments are calculated using a few terms. The reliability and simplicity of the method areillustrated with numerical examples and plots. The Laplace Adomian Decomposition algorithm is shown toyield very accurate approximate solutions using only a few iterations. Fourth-order Runge-Kutta solutionsare also compared with the solutions obtained by the Laplace decomposition scheme.Complex Valued Bipolar Metric Spaces and Fixed Point Theorems
https://www.cna-journal.com/article_174091.html
This article introduces the idea of complex valued bipolar metric space and derives some of its properties. Moreover, for complex valued bipolar metric spaces, various fixed point theorems of contravariant maps satisfying rational inequalities are proved. Additionally, the Kannan fixed point theorem and the Banach contraction principle are both generalised.Some high-Order convergence modifications of the Householder method for Nonlinear Equations
https://www.cna-journal.com/article_174092.html
One major setback of iterative methods that require higher derivatives in their iterative procedures is that of computational cost. The Householder&rsquo;s method is one of such methods that require second derivative evaluation in its procedures. To circumvent this setback, the second derivative is annihilated by estimation via the use of the interpolating polynomial and the divided difference techniques. Consequently, three new modifications of the Householder&rsquo;s method that are of two and three steps were put forward in this article. To further improve the efficiency of the modified methods, a weight function is introduced to the iterative cycle to enhance the methods convergence order. From the convergence analysis conducted on the methods, revealed that they are of fifth, ninth and tenth order convergence respectively. To test the applicability of the methods, they were applied to locate the solutions of some nonlinear equations and modeled practical problems that are nonlinear equations. From the computational experience, it was observed that the methods performed better than the compared methods that are also modifications of the Householder methods.New Common Fixed Point Theorem in Dislocated Quasi b-Metric Spaces
https://www.cna-journal.com/article_174093.html
In this article, a new common fixed point theorem for a pair of continous self mapping will be illustrated in the frame work of dislocated quasi b-metric space. The established theorem extend and generalize some well-known results inthe literature. Example is given in the support of the constructed resultOn Hardy-Rogers Type Contraction Mappings in Cone A_{b}-metric spaces
https://www.cna-journal.com/article_136011.html
In this manuscript, a generalized fixed point theorem of Hardy-Rogers type contraction is proved in cone A_{b}-metric spaces, which relaxes the contraction condition. Also, some fixed point results for different contraction mappings are given in such spaces.A monotone hybrid algorithm for maximal monotone operators and a family of generalized nonexpansive mappings
https://www.cna-journal.com/article_160209.html
In this paper, a new monotone hybrid method is introduced in the framework of Banach spaces for finding a common element of the set of zeros of a maximum monotone operator and the fixed point set of a family of generalized nonexpansive mappings. The prove is given in the framework of Banach spaces for the strong convergence of a sequence of iteration to a common element of the set of zeros of a maximum monotone operator and the fixed point set of a family of generalized nonexpansive mappings. New convergence results are obtained for resolvents of maximal monotone operators and a family of generalized nonexpansive mappings in a Banach space.Existence and stability for Ambartsumian equation with -Hilfer generalized proportional fractional derivative
https://www.cna-journal.com/article_171340.html
The main objective of this paper is to study the Ambartsumian equation in the sense of Hilfer Generalized proportional fractional derivative(HGPFD). The existence and stability properties of solution are studied. The technique used for study is fixed point theorem and Gronwall inequality. Ulam-Hyers-Rassias stability of the solution is also investigated.Convergence Analysis Monotone Hybrid Algorithms for Countable Family of Generalized Nonexpansive Mappings and Maximal Monotone Operators
https://www.cna-journal.com/article_174088.html
A new countable family of generalized nonexpansive mappings is introduced and a new monotone hybrid algorithm is presented in the framework of Banach spaces. Some new results are obtained for the class of generalized nonexpansive mappings and countable family of generalized nonexpansive mappings. The study exhibits the procedure for obtaining a common element of the zero point set of a maximal monotone operator and the newly introduced countable family of generalized nonexpansive mappings.Generalization of the Riesz-Markov-Kakutani Representation Theorem and Weak Spectral Families
https://www.cna-journal.com/article_174089.html
Firstly, we generalize some definitions such as the definitions of the weak spectral family, the solitary operator, and the construction of the functional calculus. Secondly, we prove that for a functional calculus $\Phi $ on the measurable space $\left(Z,\; \Sigma \right)$ exists a measurable space $\left(\Omega ,\mathrm{F},\mu \right)$, an operator $U\; :\; X\to L^{p} \left(\Omega ,\mathrm{F},\mu \right)$, and a continuous $\mathrm{\ast }$ -homomorphism $F\; :\; M\left(Z,\; \Sigma \right)\to M\left(\Omega ,\mathrm{F}\right)$, such that $M_{F\left(f\right)} =U^{-1} \Phi \left(f\right)U$ for all $f\in M\left(Z,\, \Sigma \right)$. Thirdly, we establish the correlation between the well-bounded operators and the weak spectral families. It has been proven that for a linear well-bounded operator $A\in L\left(X\right)$ there is a weak spectral family $\left\{E\left(\lambda \right)\in L\left(X^{*} \right),\quad \lambda \in {\mathbb R}\right\}$ on a compact interval $\left[a,\; b\right]$ such that an integral representation $\left\langle A\left(x\right),\; y^{*} \right\rangle =b\left\langle x,\; y^{*} \right\rangle -\int _{\left[a,\; b\right]}\left\langle x,\; E\left(\lambda \right)y^{*} \right\rangle d\lambda $ holds for all $x\in X$, $y^{*} \in X^{*} $, where equivalence is understood in the weak topology.Analytical Solution Of Fractional Order SIR Model Via Laplace Adomian Decomposition Method
https://www.cna-journal.com/article_138221.html
In this article, we developed the scheme for approximate solutions of fractional order SIR model of infectious diseases. We obtain desert results in the form of infinite series with the Adomian decomposition method coupled with Laplace transform. In the last section of this work, we provide the numerical discussion for the proposed model and plots those results for illustrative purposes via Matlab.Development of the Generalized C Operator: Characteristics, Inequalities and Examples
https://www.cna-journal.com/article_184813.html
Multi-dimensional integral operators play a crucial role in analysis by enabling the comprehensive evaluation and characterization of complex functions and systems in multiple dimensions. In this setting, the C operator has recently emerged with the distinct features of being nonlinear with an original integral ratio form and having non-linear properties, tractable series expansions and partial derivatives, manageable lower and upper bounds (including Cauchy-Schwarz-type and Lipschitz-type inequalities), convex characteristics, and closed-form expressions for a plethora of functions. In particular, it is connected with a wide panel of well-known integral operators, including the Laplace transform, exponential integral operator, logarithimic integral operator, etc. This article introduces a new multi-dimensional nonlinear operator that expands the capabilities of the C operator. The main novelties in its construction are: (i) a shape parameter that modulates or vanishes the ratio term of the former C operator; and (ii) a multi-dimensional multiplicative power function, also depending on a shape parameter. Due to these novelties, the generalized C operator can be viewed as a weighted version of the C operator. We mainly investigate its notable properties, such as various scale properties, tractable series expansions and partial derivatives, sharp lower and upper bounds, which include ordering comparisons with the original C operator, and convex characteristics. The generalized C operator applied to a few specific functions is provided; some of these represent novel integral results from the literature. Overall, our theoretical findings emphasize the benefits and potential of this new multi-dimensional nonlinear integral operator.Solving non-homogeneous lane-embed equation using Taylor series method
https://www.cna-journal.com/article_186466.html
This paper presents the approximate analytical solution for a lane-embedded non-homogeneous type
problem using the Taylor series method. The Taylor series method stands as a fundamental mathematical tool for solving nonlinear differential equations, and in this work, its effectiveness is demonstrated by yielding highly accurate approximate solution with minimal iterations. Achieving higher derivatives in Taylor&rsquo;s expansion is facilitated through the utilization of Maple software. Furthermore, in order to gain a comprehensive understanding of the system&rsquo;s behavior, we conduct a comparison with the closed-form equation.Analytical Solution of System of Fractional-Order Lane-Emden Type Equations
https://www.cna-journal.com/article_186468.html
Single Lane-Emden equations model several phenomena in physics and astrophysics such as isotropic continuous media, isothermal gas spheres and the thermal behaviour of a spherical cloud of gas. Systems of Lane-Emden equations appear in the modelling of several physical problems such as pattern formation,
chemical reactions, and population evolution. In this paper, a class of systems of fractional-order Lane-Emden type equations is considered using a fractional power series method. The proposed method applies generalised Cauchy product of power series to the nonlinear terms and explicit recurrence relations for the expansion coefficients of the system of series solutions are generated. The problem considered is a fractional-order version of a system of classical Lane-Emden equations studied (jointly) by B. Muatjetjeja and C.M. Khalique (First integrals for a generalized coupled Lane-Emden system, Nonlinear Anal.
Real World Appl., 12, 2011, 1202-1212); and the one
by A.M. Wazwaz (The variational iteration method for solving systems of equations of Emden-Fowler type,
Int. J. Comput. Math., 2011, 88, 3406-3415) using different methods. The illustrative examples presented generalise many problems in the literature. The solutions obtained agree excellently with the exact solutions and several published results from other methods.On the Approximate Solutions for Fractional Differential Equations with Caputo-Fabrizio Fractional Derivative
https://www.cna-journal.com/article_186469.html
In this paper, the Sumudu Daftardar-Jafari method, abbreviated as (SDJM) is established and successfully applied to obtain approximate analytical solutions for the fractional differential equations with the Caputo-Fabrizio fractional derivative (CFFD). SDJM is expressed with a combining of the Sumudu transform and the Daftardar-Jafari method. Based on this method, the solutions are considered in the form of a series. Three examples are provided to illustrate the efficiency and ease of implementation of the
proposed method. Moreover, the obtained solutions are compared with the exact solutions. We also present our results with the help of tables and figures.FIXED POINT THEOREMS IN DISLOCATED QUASI METRIC AND DISLOCATED QUASI b-METRIC SPACES
https://www.cna-journal.com/article_186470.html
In this paper, the concept of generalized contraction mapping has been used in
investigating fixed point theorems. A fixed point theorem of integral type will be
establish in the setting of dislocated quasi metric spaces using some new type of
rational contraction conditions. Moreover, a fixed point theorem will be studied in
the context of dislocated quasi b-metric spaces using the idea of generalized new
contraction conditions. Several corollaries may be deduced from our established
results, which generalize and extend several results of the literature. Also , our
investigated results may be easily used in solving equations in non-linear analysis too.Common Fixed Point Theorem For A Pair of Self Mappings In Dislocated Quasi b-Metric Space
https://www.cna-journal.com/article_186471.html
In this article, a common fixed point theorem for a pair of continuous self mapping will be illustrated in the frame work of dislocated quasi b-metric space. The established theorem extend and generalize some well-known results given by Rahman and et. al. Also, example is given in the support of the constructed result.On the geometric construction of the Jarratt-Gander's class and its relationship with other methods
https://www.cna-journal.com/article_186472.html
This paper establishes a geometric interpretation of the methods obtained from a two-point Jarratt-Gander type class and then these are compared with other kinds of methods to polynomial with two different roots. Finally, several numerical examples to observe the performance of the classic and new members of the families mentioned are presented.