Numerical Solution of 3rd order ODE Using FDM: On a Moving Surface in MHD Flow of Sisko Fluid

A Similarity group theoretical technique is used to transform the governing nonlinear partial differential equations of two dimensional MHD boundary layer flow of Sisko fluid into nonlinear ordinary differential equations. Then the resulting third order nonlinear ordinary differential equation with corresponding boundary conditions is linearised by Quasi linearization method. Numerical solution of the linearised third order ODE is obtained using Finite Difference method (FDM). Graphical presentation of the solution is given

Vector Additive Decomposition for 2D Fractional Diffusion Equation

Such physical processes as the diffusion in the environments with fractal geometry and the particles’ subdiffusion lead to the initial value problems for the nonlocal fractional order partial differential equations. These equations are the generalization of the classical integer order differential equations.  An analytical solution for fractional order differential equation with the constant coefficients is obtained in [1] by using Laplace-Fourier transform. However, nowadays many of the practical problems are described by the models with variable coefficients.  In this paper we discuss the numerical vector decomposition model which is based on a shifted version of usual Gr¨unwald finite-difference approximation [2] for the non-local fractional order operators. We prove the unconditional stability of the method for the fractional diffusion equation with Dirichlet boundary conditions. Moreover, a numerical example using a finite difference algorithm for 2D fractional order partial differential equations is also presented and compared with the exact analytical solution

Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations

The Laplace transform boundary element method for diffusion-type problems

Diffusion-type problems are described by parabolic partial differential equations; they are defined on a domain involving both time and space. The usual method of solution is to use a finite difference time-stepping process which leads to an elliptic equation in the space variable. The major drawback with the finite difference method in time is the possibility of severe stability restrictions. An alternative process is to use the Laplace transform. The transformed problem can be solved using a suitable partial differential equation solver and the solution is transformed back into the time domain using a suitable inversion process. In all practical situations a numerical inversion is required. For problems with discontinuous or periodic boundary conditions, the numerical inversion is not straightforward and we show how to overcome these difficulties. The boundary element method is a well-established technique for solving elliptic problems. One of the procedures required is the evaluation of singular integrals which arise in the solution process and a new formulation is developed to handle these integrals. For the solution of non-homogeneous equations an additional technique is required and the dual reciprocity method used in conjunction with the boundary element method provides a way forward. The Laplace transform is a linear operator and as such cannot handle non-linear terms. We address this problem by a linearisation process together with a suitable iterative scheme. We apply such a procedure to a non-linear coupled electromagnetic heating problem with electrical and thermal properties exhibiting temperature dependencies

On the eventual periodicity of fractional order dispersive wave equations using RBFS and transform

In this research work, let’s focus on the construction of numerical scheme based on radial basis functions finite difference (RBF-FD) method combined with the Laplace transform for the solution of fractional order dispersive wave equations. The numerical scheme is then applied to examine the eventual periodicity of the proposed model subject to the periodic boundary conditions. The implementation of proposed technique for high order fractional and integer type nonlinear partial differential equations (PDEs) is beneficial because this method is local in nature, therefore it yields and resulted in sparse differentiation matrices instead of full and dense matrices. Only small dimensions of linear systems of equations are to be solved for every center in the domain and hence this procedure is more reliable and efficient to solve large scale physical and engineering problems in complex domain. Laplace transform is utilized for obtaining the equivalent time-independent equation in Laplace space and also valuable to handle time-fractional derivatives in the Caputo sense. Application of Laplace transform avoids the time steeping procedure which commonly encounters the time instability issues. The solution to the transformed model is then obtained by computing the inversion of Laplace transform with an appropriate contour in a complex space, which is approximated by trapezoidal rule with high accuracy. Also since the Laplace transform operator is linear, it cannot be used to transform non-linear terms therefore let’s use a linearization approach and an appropriate iterative scheme. The proposed approach is tasted for some nonlinear fractional order KdV and Burgers equations. The capacity, high order accuracy and efficiency of our approach are demonstrated using examples and resultsRBFs Method

A parallel hybrid implementation of the 2D acoustic wave equation

In this paper, we propose a hybrid parallel programming approach for a numerical solution of a two-dimensional acoustic wave equation using an implicit difference scheme for a single computer. The calculations are carried out in an implicit finite difference scheme. First, we transform the differential equation into an implicit finite-difference equation and then using the ADI method, we split the equation into two sub-equations. Using the cyclic reduction algorithm, we calculate an approximate solution. Finally, we change this algorithm to parallelize on GPU, GPU+OpenMP, and Hybrid (GPU+OpenMP+MPI) computing platforms. The special focus is on improving the performance of the parallel algorithms to calculate the acceleration based on the execution time. We show that the code that runs on the hybrid approach gives the expected results by comparing our results to those obtained by running the same simulation on a classical processor core, CUDA, and CUDA+OpenMP implementations.Comment: 10 pages; 1 Chart; 1 Table; 1 Listing; 1 Algorith

Differential equations and asymptotic solutions for arithmetic Asian options: \u27Black-Scholes formulae\u27 for Asian rate calls

In this article, we present a simplified means of pricing Asian options using partial differential equations (PDEs). We first provide a concise derivation of the well-known similarity reduction and exact Laplace transform solution. We then analyse the problem afresh as a power series in the volatility-scaled contract duration, with a view to obtaining an asymptotic solution for the low-volatility limit, a limit which presents difficulties in the context of the general Laplace transform solution. The problem is approached anew from the point of view of asymptotic expansions and the results are compared with direct, high precision, inversion of the Laplace transform and with numerical results obtained by V. Linetsky and J. Vecer. Our asymptotic formulae are little more complicated than the standard BlackScholes formulae and, working to third order in the volatility-scaled expiry, are accurate to at least four significant figures for standard test problems. In the case of zero risk-neutral drift, we have the solution to fifth order and, for practical purposes, the results are effectively exact. We also provide comparisons with the hybrid analytic and finite-difference method of Zhang

THE NUMERICAL APPROXIMATION FOR THE INTEGRABILITY PROBLEM AND THE MEASURE OF WELFARE CHANGES, AND ITS APPLICATIONS

This dissertation mainly studied on numerical approximation methods as a solution of the integrability problem and the measure of welfare changes, and demonstrated how numerical algorithms can be applied in empirical studies as a solution method. In general, the integrability problem is described as a system of the partial differential equations (PDE) in terms of the expenditure function, and the measure of welfare changes is defined by the difference between the expenditure function at two different time periods. Both problems can be solved using the same method since solutions for these questions mainly relied on how to recover the compensated income (expenditure) from the ordinary demand function. In order to investigate whether numerical approximation methods can be applied to the integrability problem and the measure of welfare changes, first, we studied the integrability problem mainly focusing on how to transform the system of the partial differential equations to the system of the ordinary differential equation since this transform possibility provides a way to solve the integrability problem using the numerical method. Second, several numerical methods were investigated as a possible solution of both problem including the Vartia, the RK-4th order algorithm, and the Adams Fourth-Order Predictor-Corrector algorithm. In addition, the Rotterdam and Almost Ideal demand system were investigated since the demand system played an important role on recovering the expenditure. Two empirical studies are performed. In the first application, using both the U.S consumer expenditure (CE) data and the consumer price index (CPI), the AI and Rotterdam demand system were estimated, and the expenditure was recovered from the estimated demand system using numerical approximation methods. From this, we could demonstrate the power and the applicability of numerical algorithms. In the second application, we paid attention to analyze the welfare effect on the U.S elderly population when prices changed. The burden index and the compensating variation were calculated using the numerical algorithm. From the evaluation, we could confirm that the welfare changes and consumer welfare losses of the elderly population were larger than that of the general U.S populatio

Semilineaarisen lämpöyhtälön ratkaisun olemassaolo ja yksikäsitteisyys

This thesis is about the existence and uniqueness of a solution for the semilinear heat equation of polynomial type. The extensive study of properties for these equations started off in the 1960s, when Hiroshi Fujita published his results that the existence and uniqueness of solutions depends critically on the exponent of the nonlinear term. In this thesis we expose some of the basic methods used in the theory of linear, constant coefficient partial differential equations. These considerations lay out the groundwork for the main result of the thesis, which is the existence and uniqueness of a solution to the generalized heat equation. In Chapter 2 we expose the basics of functional analysis. We start off by defining Banach spaces and provide some examples of them. Then, we state the very useful Banach fixed point theorem, which guarantees the existence and uniqueness of a solution to certain types of integral equations. Next, we consider linear maps between normed spaces, with a focus on linear isomorphisms, which are linear maps preserving completeness. The isomorphisms prove to be very useful, when we consider weighted spaces. This is because for certain types of weights, we can identify the multiplication by weight with a linear isomorphism. In Chapter 3 we introduce the Fourier transform, which is a highly useful tool for studying linear partial differential equations. We go through its basic mapping properties, such as, interaction with derivatives and convolution. Then, we consider useful spaces in Fourier analysis. Chapter 4 is on the regular, inhomogeneous heat equation. A common method for deriving the solution to heat equation is formally applying the Fourier transform to it. This way we obtain a first order, linear ordinary differential equation, for which there is a known solution. The derived solution will serve as a motivator for how to approach the semilinear case. Also, in the end we will solve explicitly a slight generalization of the heat equation. In Chapter 5 we prove the main result of this thesis: existence and uniqueness of a generalized solution for the semilinear heat equation. The methods we use in the proof are quite elementary in the sense that we do not need heavy mathematical machinery. We reformulate the generalized semilinear heat equation using an operator and show that it satisfies the conditions of the Banach fixed point theorem in a small, closed ball of a suitable Banach space. We also include an appendix, in which we discuss differentiability properties of the generalized solution. It is possible to apply methods used in the proof of the generalized case to prove continuous differentiability. We provide some ideas on how one should approach the time differentiability of the solution by estimating the difference quotient of the integral operator