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Application of adomian decomposition method to solving nonlinear differential equationsSekgothe, Nkhoreng Hazel January 2021 (has links)
Thesis (M. Sc. (Applied Mathematics)) -- University of Limpopo, 2021 / Modelling with differential equations is of paramount importance as it provides pertinent insight into the dynamics of many engineering and technological devices and/or processes. Many such models, however, involve differential equations that are inherently nonlinear and difficult to solve. Many numerical methods have been developed to solve a variety of differential equations that cannot be solved analytically. Most numerical methods, however, require discretisation, linearisation of the nonlinear terms and other simplifying approximations that may inhibit the accuracy
of the solution. Further, in some methods high computational complexity is involved. Due to the importance of differential equations in modelling real life phenomena and these stated shortfalls, continuous pursuit of more efficient solution techniques by the scientific community is ongoing. Industrial and technological advancement are to a larger extent dependent upon efficient and accurate solution techniques. In this work, we investigate the use of Adomian decomposition method in solving nonlinear ordinary and partial differential equations. One advantage of Adomian decomposition method that has been demonstrated in literature is that it achieves a rapidly convergent infinite series
solution. The method is also advantageous in that it does not require one to linearise and
discretise the equations as is done with other numerical methods. In our investigation, among other important examples, we will apply the Adomian decomposition method to solve selected fluid flow and heat transfer problems. Fluid flow and heat transfer models have pertinent applications in engineering and technology. The Adomian decomposition method will be compared with other series solution methods, namely the differential transform method and the homotopy analysis method. The desirable attributes of the Adomian decomposition method that are stated in literature have been ascertained in this work and it has also been demonstrated that the Adomian decomposition method compares favourably with the other series solution methods. It has also been demonstrated that in some cases nonlinear complexity results in slow convergence rate of
the Adomian decomposition method.
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Finite element solution of the reaction-diffusion equationMahlakwana, Richard Kagisho January 2020 (has links)
Thesis (M.Sc. (Applied Mathematics)) -- University of Limpopo, 2020 / In this study we present the numerical solution o fboundary value problems for
the reaction-diffusion equations in 1-d and 2-d that model phenomena such as
kinetics and population dynamics.These differential equations are solved nu-
merically using the finite element method (FEM).The FEM was chosen due to
several desirable properties it possesses and the many advantages it has over
other numerical methods.Some of its advantages include its ability to handle
complex geometries very well and that it is built on well established Mathemat-
ical theory,and that this method solves a wider class of problems than most
numerical methods.The Lax-Milgram lemma will be used to prove the existence
and uniqueness of the finite element solutions.These solutions are compared
with the exact solutions,whenever they exist,in order to examine the accuracy
of this method.The adaptive finite element method will be used as a tool for
validating the accuracy of theFEM.The convergence of the FEM will be proven
only on the real line.
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Efficient solutions of 2-D incompressible steady laminar separated flowsMorrison, Joseph H. January 1986 (has links)
This thesis describes a simple efficient and robust numerical technique for solving two-dimensional incompressible laminar steady flows at moderate-to-high Reynolds numbers. The method uses an incremental multigrid method and an extrapolation procedure based on minimum residual concepts to accelerate the convergence rate of a robust block-line-Gauss-Seidel solver for the vorticity-stream function equations. Results are presented for the driven cavity flow problem using uniform and nonuniform grids and for the flow past a backward facing step in a channel. / M.S.
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Nondifferentiable optimization algorithms with application to solving Lagrangian dual problemsChoi, Gyunghyun 19 June 2006 (has links)
In this research effort, we consider nondifferentiable optimization (NDO) problems that arise in several applications in science, engineering, and management, as well as in the context of other mathematical programming approaches such as Dantzig-Wolfe decomposition, Benders decomposition, Lagrangian duality, penalty function methods, and minimax problems. The importance and necessity of having effective solution methods for NDO problems has long been recognized by many scientists and engineers. However, the practical use of NDO techniques has been somewhat limited, principally due to the lack of computationally viable procedures, that are also supported by theoretical convergence properties, and are suitable for solving large-scale problems. In this research, we present some new algorithms that are based on popular computationally effective strategies, while at the same time, do not compromise on theoretical convergence issues.
First, a new variable target value method (VTVM) is introduced that has an e-convergence property, and that differs from other known methods in that it does not require any prior assumption regarding bounds on an optimum or regarding the solution space. In practice, the step-length is often calculated by using an estimate of the optimal objective function value. For general nondifferentiable optimization problems, however, this may not be readily available. Hence, we design an algorithm that does not assume the possibility of having an a prior estimate of the optimal objective function value. Furthermore, along with this new step-length rule, we present a new subgradient deflection strategy in which a selected subgradient is rotated optimally toward a point that has an objective function value less than the incumbent target value. We also develop another deflection strategy based on Shor’s space dilation algorithm, so that the resulting direction of motion turns out to be a particular convex combination of two successive subgradients and we establish suitable convergence results.
In the second part of this dissertation, we consider Lagrangian dual problems. Our motivation here is the inadequacy of the simplex method or even interior point methods to obtain quick, near-optimal solutions to large linear programming relaxations of certain discrete or combinatorial optimization problems. Lagrangian dual methods, on the other hand, are quite well equipped to deal with complicating constraints and ill-conditioning problems. However, available optimization methods for such problems are not very satisfactory, and can stall far from the optimal objective function value for some problems. Also, there is no practical implementation strategy for recovering a primal optimal solution. This is a major shortcoming, even if the method is used only for bounding purposes in the context of a branch and bound scheme. With this motivation, we present new primal convergence theorems that generalize existing results on recovering primal optimal solutions, and we describe a scheme for embedding these results within a practical primal-dual Lagrangian optimization procedure. / Ph. D.
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Numerical approximation and identification problems for singular neutral equationsCerezo, Graciela M. 05 September 2009 (has links)
A collocation technique in non-polynomial spline space is presented to approximate solutions of singular neutral functional differential equations (SNFDEs). Using solution representations and general well-posedness results for SNFDEs convergence of the method is shown for a large class of initial data including the case of discontinuous initial function. Using this technique, an identification problem is solved for a particular SNFDE. The technique is also applied to other different examples. Even for the special case in which the initial data is a discontinuous function the identification problem is successfully solved. / Master of Science
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Viscous-inviscid interactions of dense gasesPark, Sang-Hyuk 11 May 2006 (has links)
The interaction of oblique shocks and oblique compression waves with a laminar boundary layer on an adiabatic flat plate is analyzed by solving the Navier-Stokes equations in conservation-law form numerically. The numerical scheme is based on the Beam and Warming’s implicit method with approximate factorization. We examine the flow of Bethe-Zel’dovich-Thompson (BZT) fluids at pressures and temperatures on the order of those of the thermodynamic critical point. A BZT fluid is a single-phase gas having specific heat so large that the fundamental derivative of gas dynamics, Γ, is negative over a finite range of pressures and temperatures. The equation of state is the well-known Martin-Hou equation. The main result is the demonstration that the natural dynamics of BZT fluids can suppress boundary layer separation. Physically, this suppression can be attributed to the decrease in adverse pressure gradients associated with the disintegration of compression discontinuities in BZT fluids. / Ph. D.
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Asymptotics of the Fredholm determinant corresponding to the first bulk critical universality class in random matrix modelsBothner, Thomas Joachim 06 November 2013 (has links)
Indiana University-Purdue University Indianapolis (IUPUI) / We study the one-parameter family of determinants $det(I-\gamma K_{PII}),\gamma\in\mathbb{R}$ of an integrable Fredholm operator $K_{PII}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $\Psi$-function associated with the Hastings-McLeod solution of the second Painlev\'e equation. In case $\gamma=1$, this Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics of $\det(I-\gamma K_)$ for all values of the real parameter $\gamma$.
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Existência de solução de equações integrais não lineares em escalas temporais sobre espaços de BanachMartins, Camila Aversa [UNESP] 27 June 2013 (has links) (PDF)
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martins_ca_me_sjrp.pdf: 297081 bytes, checksum: 4a6f13bdad08f9e72c8df07186762615 (MD5) / Neste trabalho estabelecemos condições para a existência e unicidade de solução para equações integrais do tipo Volterra–Stieltjes não linear x(t)+ Z [a,t]T DsK(t,s) f (s,x(s)) = u(t), t E [a,b]T em escalas temporais T, usando a integral de Cauchy–Stieltjes à direita sobre funções regradas a valores em espaços de Banach / In this work we establish conditions for the existence and uniqueness of solution a Volterra– Stieltjes integral nonlinear equations x(t)+ Z [a,t]T DsK(t,s) f (s,x(s)) = u(t), t E [a,b]Tin time scales T, using the right Cauchy–Stieltjes integral on regulated functions with values in Banach spaces
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The application of the multigrid algorithm to the solution of stiff ordinary differential equations resulting from partial differential equations.Parumasur, Nabendra. January 1992 (has links)
We wish to apply the newly developed multigrid method [14] to the solution of
ODEs resulting from the semi-discretization of time dependent PDEs by the
method of lines. In particular, we consider the general form of two important
PDE equations occuring in practice, viz. the nonlinear diffusion equation and
the telegraph equation. Furthermore, we briefly examine a practical area, viz.
atmospheric physics where we feel this method might be of significance. In
order to offer the method to a wider range of PC users we present a computer
program, called PDEMGS. The purpose of this program is to relieve the user
of much of the expensive and time consuming effort involved in the solution
of nonlinear PDEs. A wide variety of examples are given to demonstrate the
usefulness of the multigrid method and the versatility of PDEMGS. / Thesis (M.Sc.)-University of Natal, Durban, 1992.
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Existência de solução de equações integrais não lineares em escalas temporais sobre espaços de Banach /Martins, Camila Aversa. January 2013 (has links)
Orientador: Luciano Barbanti / Coorientador: Geraldo Nunes Silva / Banca: German Jesus Lozada Cruz / Banca: Márcia Cristina Anderson Braz Federson / Resumo: Neste trabalho estabelecemos condições para a existência e unicidade de solução para equações integrais do tipo Volterra-Stieltjes não linear x(t)+ Z [a,t]T DsK(t,s) f (s,x(s)) = u(t), t E [a,b]T em escalas temporais T, usando a integral de Cauchy-Stieltjes à direita sobre funções regradas a valores em espaços de Banach / Abstract: In this work we establish conditions for the existence and uniqueness of solution a Volterra- Stieltjes integral nonlinear equations x(t)+ Z [a,t]T DsK(t,s) f (s,x(s)) = u(t), t E [a,b]Tin time scales T, using the right Cauchy-Stieltjes integral on regulated functions with values in Banach spaces / Mestre
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