Analytical Solution of Biological Population of Fractional Differential Equations by Reconstruction of Variational Iteration Method

This article presents a brand-new approximation analytical technique we refer to as the reconstruction of variational iteration method. For the goal of solving fractional biological population option pricing equations, this methodology was created. In certain circumstances, you may actually use the well-known Mittag-Leffer function to get an explicit response. The usage of the three examples below demonstrates the precision and effectiveness of the suggested method. The results show that the RVIM is not only quite straightforward but also very successful at resolving non-linear problems

Algorithm for solving a generalized Hirota-Satsuma Coupled KdV equation using homotopy perturbation transform method

In this paper, we merge homotopy perturbation method with He’s polynomials and Laplace transformation method to produce a highly effective algorithm for finding approximate solutions for generalized Hirota-Satsuma Coupled KdV equations. This technique is called the Homotopy Perturbation Transform Method (HPTM). With this technique, the solutions are obtained without any discretization or restrictive assumptions, and devoid of roundoff errors. This technique solved a generalized Hirota-Satsuma Coupled KdV equation without using Adomian’s polynomials which can be considered as a clear advantage over the decomposition method. MAPLE software was used to calculate the series generated from the algorithm. The results reveal that the homotopy perturbation transform method (HPTM) is very efficient, simple and can be applied to other nonlinear problems.Keywords: Coupled KdV equations, Homotopy perturbation transform method, Laplace transform method, Maple software, He’s polynomia

Two Hybrid Methods for Solving Two-Dimensional Linear Time-Fractional Partial Differential Equations

A computationally efficient hybridization of the Laplace transform with two spatial discretization techniques is investigated for numerical solutions of time-fractional linear partial differential equations in two space variables. The Chebyshev collocation method is compared with the standard finite difference spatial discretization and the absolute error is obtained for several test problems. Accurate numerical solutions are achieved in the Chebyshev collocation method subject to both Dirichlet and Neumann boundary conditions. The solution obtained by these hybrid methods allows for the evaluation at any point in time without the need for time-marching to a particular point in time

On the application of partial differential equations and fractional partial differential equations to images and their methods of solution

This body of work examines the plausibility of applying partial di erential equations and time-fractional partial di erential equations to images. The standard di usion equation is coupled with a nonlinear cubic source term of the Fitzhugh-Nagumo type to obtain a model with di usive properties and a binarizing e ect due to the source term. We examine the e ects of applying this model to a class of images known as document images; images that largely comprise text. The e ects of this model result in a binarization process that is competitive with the state-of-the-art techniques. Further to this application, we provide a stability analysis of the method as well as high-performance implementation on general purpose graphical processing units. The model is extended to include time derivatives to a fractional order which a ords us another degree of control over this process and the nature of the fractionality is discussed indicating the change in dynamics brought about by this generalization. We apply a semi-discrete method derived by hybridizing the Laplace transform and two discretization methods: nite-di erences and Chebyshev collocation. These hybrid techniques are coupled with a quasi-linearization process to allow for the application of the Laplace transform, a linear operator, to a nonlinear equation of fractional order in the temporal domain. A thorough analysis of these methods is provided giving rise to conditions for solvability. The merits and demerits of the methods are discussed indicating the appropriateness of each method

A finite volume approach for the numerical analysis and solution of the Buckley-Leverett equation including capillary pressure

The study of petroleum recovery is significant for reservoir engineers. Mathematical models of the immiscible displacement process contain various assumptions and parameters, resulting in nonlinear governing equations which are tough to solve. The Buckley-Leverett equation is one such model, where controlling forces like gravity and capillary forces directly act on saturation profiles. These saturation profiles have important features during oil recovery. In this thesis, the Buckley-Leverett equation is solved through a finite volume scheme, and capillary forces are considered during this calculation. The detailed derivation and calculation are also illustrated here. First, the method of characteristics is used to calculate the shock speed and characteristics curve behaviour of the Buckley-Leverett equation without capillary forces. After that, the local Lax-Friedrichs finite-volume scheme is applied to the governing equation (assuming there are no capillary and gravity forces). This mathematical formulation is used for the next calculation, where the cell-centred finite volume scheme is applied to the Buckley- Leverett equation including capillary forces. All calculations are performed in MATLAB. The fidelity is also checked when the finite-volume scheme is computed in the case where an analytical solution is known. Without capillary pressure, all numerical solutions are calculated using explicit methods and smaller time steps are used for stability. Later, the fixed-point iteration method is followed to enable the stability of the local Lax-Friedrichs and Cell-centred finite volume schemes using an implicit formulation. Here, we capture the number of iterations per time-steps (including maximum and average iterations per time-step) to get the solution of water saturation for a new time-step and obtain the saturation profile. The cumulative oil production is calculated for this study and illustrates capillary effects. The influence of viscosity ratio and permeability in capillary effects is also tested in this study. Finally, we run a case study with valid field data and check every calculation to highlight that our proposed numerical schemes can capture capillary pressure effects by generating shock waves and providing single-valued saturation at each position. These saturation profiles help find the amount of water needed in an injection well to displace oil through a production well and obtains good recovery using the water flooding technique

Modeling and inversion of seismic data using multiple scattering, renormalization and homotopy methods

Seismic scattering theory plays an important role in seismic forward modeling and is the theoretical foundation for various seismic imaging methods. Full waveform inversion is a powerful technique for obtaining a high-resolution model of the subsurface. One objective of this thesis is to develop convergent scattering series solutions of the Lippmann-Schwinger equation in strongly scattering media using renormalization and homotopy methods. Other objectives of this thesis are to develop efficient full waveform inversion methods of time-lapse seismic data and, to investigate uncertainty quantification in full waveform inversion for anisotropic elastic media based on integral equation approaches and the iterated extended Kalman filter. The conventional Born scattering series is obtained by expanding the Lippmann-Schwinger equation in terms of an iterative solution based on perturbation theory. Such an expansion assumes weak scattering and may have the problems of convergence in strongly scattering media. This thesis presents two scattering series, referred to as convergent Born series (CBS) and homotopy analysis method (HAM) scattering series for frequency-domain seismic wave modeling. For the convergent Born series, a physical interpretation from the renormalization prospective is given. The homotopy scattering series is derived by using homotopy analysis method, which is based on a convergence control parameter $h$ and a convergence control operator $H$ that one can use to ensure convergence for strongly scattering media. The homotopy scattering scattering series solutions of the Lippmann-Schwinger equation, which is convergent in strongly scattering media. The homotopy scattering series is a kind of unified scattering series theory that includes the conventional and convergent Born series as special cases. The Fast Fourier Transform (FFT) is employed for efficient implementation of matrix-vector multiplication for the convergent Born series and the homotopy scattering series. This thesis presents homotopy methods for ray based seismic modeling in strongly anisotropic media. To overcome several limitations of small perturbations and weak anisotropy in obtaining the traveltime approximations in anisotropic media by expanding the anisotropic eikonal equation in terms of the anisotropic parameters and the elliptically anisotropic eikonal equation based on perturbation theory, this study applies the homotopy analysis method to the eikonal equation. Then this thesis presents a retrieved zero-order deformation equation that creates a map from the anisotropic eikonal equation to a linearized partial differential equation system. The new traveltime approximations are derived by using the linear and nonlinear operators in the retrieved zero-order deformation equation. Flexibility on variable anisotropy parameters is naturally incorporated into the linear differential equations, allowing a medium of arbitrarily anisotropy. This thesis investigates efficient target-oriented inversion strategies for improving full waveform inversion of time-lapse seismic data based on extending the distorted Born iterative T-matrix inverse scattering to a local inversion of a small region of interest (e. g. reservoir under production). The target-oriented approach is more efficient for inverting the monitor data. The target-oriented inversion strategy requires properly specifying the wavefield extrapolation operators in the integral equation formulation. By employing the T-matrix and the Gaussian beam based Green’s function, the wavefield extrapolation for the time-lapse inversion is performed in the baseline model from the survey surface to the target region. I demonstrate the method by presenting numerical examples illustrating the sequential and double difference strategies. To quantify the uncertainty and multiparameter trade-off in the full waveform inversion for anisotropic elastic media, this study applies the iterated extended Kalman filter to anisotropic elastic full waveform inversion based on the integral equation method. The sensitivity matrix is an explicit representation with Green’s functions based on the nonlinear inverse scattering theory. Taking the similarity of sequential strategy between the multi-scale frequency domain full waveform inversion and data assimilation with an iterated extended Kalman filter, this study applies the explicit representation of sensitivity matrix to the the framework of Bayesian inference and then estimate the uncertainties in the full waveform inversion. This thesis gives results of numerical tests with examples for anisotropic elastic media. They show that the proposed Bayesian inversion method can provide reasonable reconstruction results for the elastic coefficients of the stiffness tensor and the framework is suitable for accessing the uncertainties and analysis of parameter trade-offs

Connectionist Learning Based Numerical Solution of Differential Equations

It is well known that the differential equations are back bone of different physical systems. Many real world problems of science and engineering may be modeled by various ordinary or partial differential equations. These differential equations may be solved by different approximate methods such as Euler, Runge-Kutta, predictor-corrector, finite difference, finite element, boundary element and other numerical techniques when the problems cannot be solved by exact/analytical methods. Although these methods provide good approximations to the solution, they require a discretization of the domain via meshing, which may be challenging in two or higher dimension problems. These procedures provide solutions at the pre-defined points and computational complexity increases with the number of sampling points.In recent decades, various machine intelligence methods in particular connectionist learning or Artificial Neural Network (ANN) models are being used to solve a variety of real-world problems because of its excellent learning capacity. Recently, a lot of attention has been given to use ANN for solving differential equations. The approximate solution of differential equations by ANN is found to be advantageous but it depends upon the ANN model that one considers. Here our target is to solve ordinary as well as partial differential equations using ANN. The approximate solution of differential equations by ANN method has various inherent benefits in comparison with other numerical methods such as (i) the approximate solution is differentiable in the given domain, (ii) computational complexity does not increase considerably with the increase in number of sampling points and dimension of the problem, (iii) it can be applied to solve linear as well as non linear Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). Moreover, the traditional numerical methods are usually iterative in nature, where we fix the step size before the start of the computation. After the solution is obtained, if we want to know the solution in between steps then again the procedure is to be repeated from initial stage. ANN may be one of the ways where we may overcome this repetition of iterations. Also, we may use it as a black box to get numerical results at any arbitrary point in the domain after training of the model.Few authors have solved ordinary and partial differential equations by combining the feed forward neural network and optimization technique. As said above that the objective of this thesis is to solve various types of ODEs and PDEs using efficient neural network. Algorithms are developed where no desired values are known and the output of the model can be generated by training only. The architectures of the existing neural models are usually problem dependent and the number of nodes etc. are taken by trial and error method. Also, the training depends upon the weights of the connecting nodes. In general, these weights are taken as random number which dictates the training. In this investigation, firstly a new method viz. Regression Based Neural Network (RBNN) has been developed to handle differential equations. In RBNN model, the number of nodes in hidden layer may be fixed by using the regression method. For this, the input and output data are fitted first with various degree polynomials using regression analysis and the coefficients involved are taken as initial weights to start with the neural training. Fixing of the hidden nodes depends upon the degree of the polynomial.We have considered RBNN model for solving different ODEs with initial/boundary conditions. Feed forward neural model and unsupervised error back propagation algorithm have been used for minimizing the error function and modification of the parameters (weights and biases) without use of any optimization technique. Next, single layer Functional Link Artificial Neural Network (FLANN) architecture has been developed for solving differential equations for the first time. In FLANN, the hidden layer is replaced by a functional expansion block for enhancement of the input patterns using orthogonal polynomials such as Chebyshev, Legendre, Hermite, etc. The computations become efficient because the procedure does not need to have hidden layer. Thus, the numbers of network parameters are less than the traditional ANN model. Varieties of differential equations are solved here using the above mentioned methods to show the reliability, powerfulness, and easy computer implementation of the methods. As such singular nonlinear initial value problems such as Lane-Emden and Emden-Fowler type equations have been solved using Chebyshev Neural Network (ChNN) model. Single layer Legendre Neural Network (LeNN) model has also been developed to handle Lane-Emden equation, Boundary Value Problem (BVP) and system of coupled ordinary differential equations. Unforced Duffing oscillator and unforced Van der Pol-Duffing oscillator equations are solved by developing Simple Orthogonal Polynomial based Neural Network (SOPNN) model. Further, Hermite Neural Network (HeNN) model is proposed to handle the Van der Pol-Duffing oscillator equation. Finally, a single layer Chebyshev Neural Network (ChNN) model has also been implemented to solve partial differential equations

Advances in Vibration Analysis Research

Vibrations are extremely important in all areas of human activities, for all sciences, technologies and industrial applications. Sometimes these Vibrations are useful but other times they are undesirable. In any case, understanding and analysis of vibrations are crucial. This book reports on the state of the art research and development findings on this very broad matter through 22 original and innovative research studies exhibiting various investigation directions. The present book is a result of contributions of experts from international scientific community working in different aspects of vibration analysis. The text is addressed not only to researchers, but also to professional engineers, students and other experts in a variety of disciplines, both academic and industrial seeking to gain a better understanding of what has been done in the field recently, and what kind of open problems are in this area

Extracting a general iterative method from an Adomian decomposition method and comparing it to the variational iteration method

AbstractIn this work, a new form of Adomian decomposition method (ADM) is presented; by this form a general iterative method can be achieved in which there is no need of calculating Adomian polynomials. Also, this general iterative method is compared with the Adomian decomposition method and variational iteration method (VIM) and its advantages are expressed