THE ENHANCED HOMOTOPY PERTURBATION METHOD FOR AXIAL VIBRATION OF STRINGS

A governing equation is established for string axial vibrations with temporal and spatial damping forces by the Hamilton principle. It is an extension of the well-known Klein-Gordon equation. The classical homotopy perturbation method (HPM) fails to analyze this equation, and a modification with an exponential decay parameter is suggested. The analysis shows that the amplitude behaves as an exponential decay by the damping parameter. Furthermore, the frequency equation is established and the stability condition is performed. The modified homotopy perturbation method yields a more effective result for the nonlinear oscillators and helps to overcome the shortcoming of the classical approach. The comparison between the analytical solution and the numerical solution shows an excellent agreement

Some Applications of Homotopy Perturbation Method

In this thesis paper, I review the basic idea of Homotopy perturbation method (HPM), Modified Homotopy perturbation method (MHPM) and Homotopy perturbation transform method (HPTM). Then apply these on some higher order non-linear problems.Further, I tried to compare the results obtained from Modified homotopy perturbation method with HPM using the Sine-Gordon and fractional Klein-Gordon equation respectively. Homotopy perturbation transform method is the coupling of homotopy perturbation and Laplace transform method. Lastly, I applied the homotopy perturbation and homotopy perturbation transform method for solving linear and non-linear Schrödinger equation

Splitting Decomposition Homotopy Perturbation Method To Solve One -Dimensional Navier -Stokes Equation

We have proposed in this  research a new scheme to find analytical  approximating solutions for Navier-Stokes equation  of  one  dimension. The  new  methodology depends on combining  Adomian  decomposition  and Homotopy perturbation methods  with the splitting time scheme for differential operators . The new methodology is applied on two problems of  the test: The first has an exact solution  while  the other one has no  exact solution. The numerical results we  obtained  from solutions of two problems, have good convergent  and high  accuracy   in comparison with the two traditional Adomian  decomposition  and Homotopy  perturbationmethods .&nbsp

Semi-analytical and numerical study of fractal fractional nonlinear system under Caputo fractional derivative

The article aims to investigate the fractional Drinfeld-Sokolov-Wilson system with fractal dimensions under the power-law kernel. The integral transform with the Adomian decomposition technique is applied to investigate the general series solution as well as study the applications of the considered model with fractal-fractional dimensions. For validity, a numerical case with appropriate subsidiary conditions is considered with a detailed numerical/physical interpretation. The absolute error in the considered exact and obtained series solutions is also presented. From the obtained results, it is revealed that minimizing the fractal dimension reinforces the amplitude of the solitary wave solution. Moreover, one can see that reducing the fractional order α marginally reduces the amplitude as well as alters the nature of the solitonic waves. It is also revealed that for insignificant values of time, solutions of the coupled system in the form of solitary waves are in good agreement. However, when one of the parameters (fractal/fractional) is one and time increases, the amplitude of the system also increases. From the error analysis, it is noted that the absolute error in the solutions reduces rapidly when x enlarges at small-time t, whereas, increment in iterations decreases error in the system. Finally, the results show that the considered method is a significant mathematical approach for studying linear/nonlinear FPDE's and therefore can be extensively applied to other physical models

Some classical integrable systems with topological solitons

This thesis is concerned with some low dimensional non-linear systems of partial differential equations and their solutions. The systems are all in the classical domain and aside from a version of one model in Appendix D, are continuous. To begin with we examine the field equations of motion derived from Hamiltonian and Lagrangian densities, respectively defining the (1 + 1)-dimensional Hyperbolic Heisenberg and Hyperbolic sigma models, where the metric on the target manifold is indefinite. The models are integrable in the sense that a suitable Lax pair exists, and admit solitonic solutions classifiable by an integer winding number. Such solutions are explicitly derived in both the static and time dependent cases where physical space X is the circle S(^1). The existence of travelling wave solutions of topological type is discussed for each model with X = S(^1) and X = R; explicit solutions are derived for the X = S(^1) case and it is shown for both the Heisenberg and sigma models, that no such travelling wave solutions exist if X is the real line. Nevertheless, time dependent solutions (not of travelling wave type) are possible in each case for X = R, some examples of which are derived explicitly. A further integrable system; the Hyperbolic 'Pivotal' model is proposed as a special case of a more general model on Hermitian symmetric spaces. Of particular interest is the fact that the Pivotal model interpolates between the previous two models. To begin with the integrability of the model is established via a Lax representation. Solutions analogous to some of those of the previous models are then derived and the interpolative limits examined with respect to the Heisenberg and sigma models. Conserved currents for the model are also briefly discussed. Finally, some conclusions and further possibilities are noted including a brief examination of a discrete version of the sigma model where the target manifold is positive definite. A Bogomol'nyi bound is shown to exist for the systems energy in terms of a well defined winding number

On new and improved semi-numerical techniques for solving nonlinear fluid flow problems.

Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.Most real world phenomena is modeled by ordinary and/or partial differential equations. Most of these equations are highly nonlinear and exact solutions are not always possible. Exact solutions always give a good account of the physical nature of the phenomena modeled. However, existing analytical methods can only handle a limited range of these equations. Semi-numerical and numerical methods give approximate solutions where exact solutions are impossible to find. However, some common numerical methods give low accuracy and may lack stability. In general, the character and qualitative behaviour of the solutions may not always be fully revealed by numerical approximations, hence the need for improved semi-numerical methods that are accurate, computational efficient and robust. In this study we introduce innovative techniques for finding solutions of highly nonlinear coupled boundary value problems. These techniques aim to combine the strengths of both analytical and numerical methods to produce efficient hybrid algorithms. In this work, the homotopy analysis method is blended with spectral methods to improve its accuracy. Spectral methods are well known for their high levels of accuracy. The new spectral homotopy analysis method is further improved by using a more accurate initial approximation to accelerate convergence. Furthermore, a quasi-linearisation technique is introduced in which spectral methods are used to solve the linearised equations. The new techniques were used to solve mathematical models in fluid dynamics. The thesis comprises of an introductory Chapter that gives an overview of common numerical methods currently in use. In Chapter 2 we give an overview of the methods used in this work. The methods are used in Chapter 3 to solve the nonlinear equation governing two-dimensional squeezing flow of a viscous fluid between two approaching parallel plates and the steady laminar flow of a third grade fluid with heat transfer through a flat channel. In Chapter 4 the methods were used to find solutions of the laminar heat transfer problem in a rotating disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation and the classical von Kάrmάn equations for boundary layer flow induced by a rotating disk. In Chapter 5 solutions of steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain bounded by two permeable surfaces and the MHD viscous flow problem due to a shrinking sheet with a chemical reaction, were solved using the new methods

New Challenges Arising in Engineering Problems with Fractional and Integer Order

Mathematical models have been frequently studied in recent decades, in order to obtain the deeper properties of real-world problems. In particular, if these problems, such as finance, soliton theory and health problems, as well as problems arising in applied science and so on, affect humans from all over the world, studying such problems is inevitable. In this sense, the first step in understanding such problems is the mathematical forms. This comes from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research done involving fractional ordinary or partial differential equations and other relevant topics relating to integer order have attracted the attention of experts from all over the world. Various methods have been presented and developed to solve such models numerically and analytically. Extracted results are generally in the form of numerical solutions, analytical solutions, approximate solutions and periodic properties. With the help of newly developed computational systems, experts have investigated and modeled such problems. Moreover, their graphical simulations have also been presented in the literature. Their graphical simulations, such as 2D, 3D and contour figures, have also been investigated to obtain more and deeper properties of the real world problem