Mathematical Sciences

We present an effective algorithm to solve the Linear and Nonlinear Klein-Gordon equation, which is based on the Perturbation Iteration Transform Method (PITM).
The Klein-Gordon equation is the name given to the equation of motion of a quantum scalar or pseudo scalar field, a field whose quanta are spin-less particles. It describes the quantum amplitude for finding a point particle in various places, the relativistic wave function, but the particle propagates both forwards and backwards in time. The Perturbation Iteration Transform Method (PITM) is a combined form of the Laplace Transform Method and Perturbation Iteration Algorithm. The method provides the solution in the form of a rapidly convergent series. Some numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. The results show that the PITM
is very efficient, simple and can be applied to other nonlinear problems.

Bloch system is mainly employed with attainment of the basic explanation of relaxation T1 and T2 and nuclear magnetic resonance (NMR). This equation is said to be the heart of both magnetic resonance imaging (MRI) and NMR spectroscopy. The major goal of this paper is to obtain an analytical solution of Fractional Nuclear Magnetic Resonance (NMR) flow equations. In this study, the use of Laplace transform and the Pade approximation has been ascertained to handle the truncated series that were obtained through the Perturbation Iteration Method for further advancement of the approximation. The fractional derivatives in this work are understood to be in the Caputo sense. The results achieved through this paper will prove that the algorithm is fit to be used for more general kinds
of fractional differential equations.

In the paper, we begin by introducing the origin of fractional calculus and the consequent application
of the Elzaki transform on fractional derivatives. The Elzaki transformation may be used to solve mathematical problems without resorting to a new frequency domain. Once we establish this connection firmly in the general setting, we turn our attention to the application of the Elzaki transform method to some non-homogeneous fractional, ordinary differential equations. Ultimately, we acquire the graphical solution of the problem by using Matlab 2013a, developed by MathWorks

Another version of the classic Sumudu Transform called the Elzaki Transform, was put forward as closely
related to the Laplace Transform. In the following paper, the Elzaki Transform Algorithm, which has been built
on the Decomposition Method, is presented to be applied to find approximate solution of a class of non-linear, initial value problems. This method gives an approximate solution in a Convergent-Series form with easily computable components necessitating no linearization or a low perturbation criterion. The most important part of this paper is the error analysis conducted between exact solutions and pade approximate solutions; it proves that our approximate solutions narrow in rapidly to the exact solutions. Moreover, as we will discuss after the results
are resented, this algorithm can also be applicable to more general classes of linear and nonlinear differential
equations.

In the following study, a novel approach called the Perturbation Iteration Algorithm PIA has been proposed and subsequently adopted for deriving and solving the Riccati differential equation. This new Perturbation Iteration Method is efficient and has no requirement
of a small parameter assumption as its earlier classical
counterparts do. Some examples have been presented to exhibit how simply and efficiently the proposed method works. After deriving the exact solution of the Riccati equation, the capability and the simplicity of the proposed technique is clarified. A percentage error for each example has also been presented.

The prime focus is on a Van der Pol-Duffing oscillator in this paper. A newly proposed method, namely; the Perturbation Iteration Algorithm (PIA) and an Alternative Variation Iteration Method (AVIM) is used to solve governing equations. The study outlines the significant features of the two methods. The beauty of the paper
lies in the error analysis between exact solutions and approximate solutions obtained by these two methods which proves that approximate solutions obtained by Alternative Variation Iteration Method converge very rapidly to the exact solutions. Both methods provide
analytical solution in the form of a convergent series with components that are easily computable, requiring no linearization or small perturbation.

Nuclear magnetic resonance flow equations, also known as the Bloch system, are said to be at the heart of both magnetic resonance imaging (MRI) and nuclear magnetic resonance (NMR) spectroscopy. The main aim of this research was to solve fractional nuclear magnetic resonance flow equations (FNMRFEs) through a numerical approach that is very easy to handle. We present a New Iterative Predictor-Corrector Algorithm (NIPCA) based on the New Iterative Algorithm and Predictor-Corrector Algorithm to solve nonlinear nuclear magnetic resonance flow equations of fractional order involving Caputo derivatives. Graphical representation of the solutions with detailed error analysis shows the higher accuracy of the new technique. This New Iterative Predictor-Corrector Algorithm requires less computational time than previously published numerical methods. The results achieved in this article indicate that the algorithm is fit to use for other chaotic systems of fractional differential equations.

Symmetry performs an essential function in finding the correct techniques for solutions to time space fractional differential equations (TSFDEs). In this article, we present the Novel Analytic Method (NAM) for approximate solutions of the linear and non-linear KdV equation for TSFDs. To enunciate the non-integer derivative for the aforementioned equation, the Caputo operator is manipulated. Furthermore, the formula implemented is a numerical way that is postulated from Taylor’s series, which confirms an analytical answer in the form of a convergent series. For delineation of the efficiency and functionality of the method in question, four applications are exemplified along with graphical interpretation and numerical solutions to finitely illustrate the behavior of the solution to this equation. Moreover, the 3D graphs of some of these numerical examples are plotted with specific values. Observing the effectiveness of this process, we can easily decide that this process can be implemented to other TSFDEs applied in the mathematical modeling of a real-world aspect.

Symmetry performs an essential function in finding the correct techniques for solutions to time space fractional differential equations (TSFDEs). In this article, we present the Novel Analytic Method (NAM) for approximate solutions of the linear and non-linear KdV equation for TSFDs. To enunciate the non-integer derivative for the aforementioned equation, the Caputo operator is manipulated. Furthermore, the formula implemented is a numerical way that is postulated from Taylor’s series, which confirms an analytical answer in the form of a convergent series. For delineation of the efficiency and functionality of the method in question, four applications are exemplified along with graphical interpretation and numerical solutions to finitely illustrate the behavior of the solution to this equation. Moreover, the 3D graphs of some of these numerical examples are plotted with specific values. Observing the effectiveness of this process, we can easily decide that this process can be implem...

In the following study, a novel approach called the Perturbation Iteration Algorithm PIA has been proposed and subsequently adopted for deriving and solving the Riccati differential equation. This new Perturbation Iteration Method is efficient and has no re-quirement of a small parameter assumption as its earlier classical counterparts do. Some examples have been presented to exhibit how simply and efficiently the proposed method works. After de-riving the exact solution of the Riccati equation, the capability and the simplicity of the proposed technique is clarified. A percentage error for each example has also been presented. Keywords:

In the paper, we begin by introducing the origin of fractional calculus and the consequent application of the Elzaki transform on fractional derivatives. The Elzaki transformation may be used to solve mathematical problems without resorting to a new frequency domain. Once we establish this connection firmly in the general setting, we turn our attention to the application of the Elzaki transform method to some non-homogeneous fractional, ordinary differential equations. Ultimately, we acquire the graphical solution of the problem by using Matlab 2013a, developed by MathWorks

Another version of the classic Sumudu Transform called the Elzaki Transform, was put forward as closely related to the Laplace Transform. In the following paper, the Elzaki Transform Algorithm, which has been built on the Decomposition Method, is presented to be applied to find approximate solution of a class of non-linear, initial value problems. This method gives an approximate solution in a Convergent-Series form with easily computable components necessitating no linearization or a low perturbation criterion. The most important part of this paper is the error analysis conducted between exact solutions and pade approximate solutions; it proves that our approximate solutions narrow in rapidly to the exact solutions. Moreover, as we will discuss after the results are resented, this algorithm can also be applicable to more general classes of linear and nonlinear differential equations.

The prime focus is on a Van der Pol-Duffing oscillator in this pa-per. A newly proposed method, namely; the Perturbation Iteration Algorithm (PIA) and an Alternative Variation Iteration Method (AVIM) is used to solve governing equations. The study outlines the significant features of the two methods. The beauty of the paper lies in the error analysis between exact solutions and approximate solutions obtained by these two methods which proves that approx-imate solutions obtained by Alternative Variation Iteration Method converge very rapidly to the exact solutions. Both methods provide analytical solution in the form of a convergent series with compo-nents that are easily computable, requiring no linearization or small perturbation.