Since its introduction in the 1980's, the Adomian Decomposition Method (ADM) has proven to be an efficient and reliable method for solving many types of problems. Originally developed to solve nonlinear functional equations, the ADM has since been used for a wide range of equation types (like boundary value problems, integral equations, equations arising in flow of incompressible and compressible fluids etc...). This work is devoted to an evaluation of the effectiveness of this method when used for fluid dynamic applications. In particular, the ADM has been applied to the Blasius equation, the Falkner-Skan equation, and the Orr-Sommerfeld equation. This study is divided into five Chapters and an Appendix. The first chapter is devoted to an introduction of the Adomian Decomposition method (ADM) with simple illustrations. The Second Chapter is devoted to the application of the ADM to generalized Blasius Equation and our result is compared to other published results when the parameter values are appropriately set. Chapter 3 presents the solution generated for the Falkner-Skan equation. Finally, the Orr-Sommerfeld equation is dealt with in the fourth Chapter. Chapter 5 is devoted to the findings and recommendations based on this study. The Appendix contains details of the solutions considered as well as an alternate solution for the generalized Blasius Equation using Bender's delta-perturbation method.

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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Nesta contribuição apresentamos uma solução para a equação de transferência radiativa condutiva em geometria cilíndrica. Esta solução é aplicada para simular a radiação e campo de temperatura juntamente com o transporte de energia radiativa e condutiva proveniente do escape liberado em lançamentos de foguetes. Para este fim, discutimos uma abordagem semianalítica reduzindo a equação original, que é contínua nas variáveis angulares, numa equação semelhante ao problema SN da transferência radiativa condutiva. A solução é construída usando um método de composição por transformada de Laplace e o método da decomposição de Adomian. O esquema recursivo ´e apresentado para o sistema de equações de ordenadas duplamente discretas juntamente com as dependências dos parâmetros e suas influências sobre a convergência heurística da solução. A solução obtida, em seguida, permite construir o campo próximo relevante para caracterizar o termo fonte para problemas de dispersão ao ajustar os parâmetros do modelo, tais como, emissividade, refletividade, albedo e outros, em comparação com a observação, que são relevantes para os processos de dispersão de campo distante e podem ser manipulados de forma independente do presente problema. Além do método de solução, também relatamos sobre algumas soluções e simulações numéricas. / In this contribution we present a solution for the radiative conductive transfer equation in cylinder geometry. This solution is applied to simulate the radiation and temperature field together with conductive and radiative energy transport originated from the exhaust released in rocket launches. To this end we discuss a semi-analytical approach reducing the original equation, which is continuous in the angular variables, into an equation similar to the SN radiative conductive transfer problem. The solution is constructed using a composite method by Laplace transform and Adomian decomposition method. The recursive scheme is presented for the doubly discrete ordinate equations system together with parameter dependencies and their influence on heuristic convergence of the solution. The obtained solution allows then to construct the relevant near field to characterize the source term for dispersion problems when adjusting the model parameters such as emissivity, reflectivity, albedo and others in comparison to the observation, that are relevant for far field dispersion processes and may be handled independently from the present problem. In addition to the solution method we also report some solutions and numerical simulations.

Nesta contribuição apresentamos uma solução para a equação de transferência radiativa condutiva em geometria cilíndrica. Esta solução é aplicada para simular a radiação e campo de temperatura juntamente com o transporte de energia radiativa e condutiva proveniente do escape liberado em lançamentos de foguetes. Para este fim, discutimos uma abordagem semianalítica reduzindo a equação original, que é contínua nas variáveis angulares, numa equação semelhante ao problema SN da transferência radiativa condutiva. A solução é construída usando um método de composição por transformada de Laplace e o método da decomposição de Adomian. O esquema recursivo ´e apresentado para o sistema de equações de ordenadas duplamente discretas juntamente com as dependências dos parâmetros e suas influências sobre a convergência heurística da solução. A solução obtida, em seguida, permite construir o campo próximo relevante para caracterizar o termo fonte para problemas de dispersão ao ajustar os parâmetros do modelo, tais como, emissividade, refletividade, albedo e outros, em comparação com a observação, que são relevantes para os processos de dispersão de campo distante e podem ser manipulados de forma independente do presente problema. Além do método de solução, também relatamos sobre algumas soluções e simulações numéricas. / In this contribution we present a solution for the radiative conductive transfer equation in cylinder geometry. This solution is applied to simulate the radiation and temperature field together with conductive and radiative energy transport originated from the exhaust released in rocket launches. To this end we discuss a semi-analytical approach reducing the original equation, which is continuous in the angular variables, into an equation similar to the SN radiative conductive transfer problem. The solution is constructed using a composite method by Laplace transform and Adomian decomposition method. The recursive scheme is presented for the doubly discrete ordinate equations system together with parameter dependencies and their influence on heuristic convergence of the solution. The obtained solution allows then to construct the relevant near field to characterize the source term for dispersion problems when adjusting the model parameters such as emissivity, reflectivity, albedo and others in comparison to the observation, that are relevant for far field dispersion processes and may be handled independently from the present problem. In addition to the solution method we also report some solutions and numerical simulations.

Nesta contribuição apresentamos uma solução para a equação de transferência radiativa condutiva em geometria cilíndrica. Esta solução é aplicada para simular a radiação e campo de temperatura juntamente com o transporte de energia radiativa e condutiva proveniente do escape liberado em lançamentos de foguetes. Para este fim, discutimos uma abordagem semianalítica reduzindo a equação original, que é contínua nas variáveis angulares, numa equação semelhante ao problema SN da transferência radiativa condutiva. A solução é construída usando um método de composição por transformada de Laplace e o método da decomposição de Adomian. O esquema recursivo ´e apresentado para o sistema de equações de ordenadas duplamente discretas juntamente com as dependências dos parâmetros e suas influências sobre a convergência heurística da solução. A solução obtida, em seguida, permite construir o campo próximo relevante para caracterizar o termo fonte para problemas de dispersão ao ajustar os parâmetros do modelo, tais como, emissividade, refletividade, albedo e outros, em comparação com a observação, que são relevantes para os processos de dispersão de campo distante e podem ser manipulados de forma independente do presente problema. Além do método de solução, também relatamos sobre algumas soluções e simulações numéricas. / In this contribution we present a solution for the radiative conductive transfer equation in cylinder geometry. This solution is applied to simulate the radiation and temperature field together with conductive and radiative energy transport originated from the exhaust released in rocket launches. To this end we discuss a semi-analytical approach reducing the original equation, which is continuous in the angular variables, into an equation similar to the SN radiative conductive transfer problem. The solution is constructed using a composite method by Laplace transform and Adomian decomposition method. The recursive scheme is presented for the doubly discrete ordinate equations system together with parameter dependencies and their influence on heuristic convergence of the solution. The obtained solution allows then to construct the relevant near field to characterize the source term for dispersion problems when adjusting the model parameters such as emissivity, reflectivity, albedo and others in comparison to the observation, that are relevant for far field dispersion processes and may be handled independently from the present problem. In addition to the solution method we also report some solutions and numerical simulations.

This master's thesis deals with solving fractional-order ordinary differential equations by the Adomian decomposition method. A part of the work is therefore devoted to the theory of equations containing differential operators of non-integer order, especially the Caputo operator. The next part is devoted to the Adomian decomposition method itself, its properties and implementation in the case of Chen system. The work also deals with bifurcation analysis of this system, both for integer and non-integer case. One of the objectives is to clarify the discrepancy in the literature concerning the fractional-order Chen system, where experiments based on the use of the Adomian decomposition method give different results for certain input parameters compared with numerical methods. The clarification of this discrepancy is based on recent theoretical knowledge in the field of fractional-order differential equations and their systems. The conclusions are supported by numerical experiments, own code implementing the Adomian decomposition method on the Chen system was used.

In this dissertation, we use the Natural transform decomposition method to obtain approximate analytical solution of fractional differential equations. This technique is a combination of decomposition methods and natural transform method. We use the Adomian decomposition, the homotopy perturbation and the Daftardar-Jafari methods as our decomposition methods. The fractional derivatives are considered in the Caputo and Caputo- Fabrizio sense. / Mathematical Sciences / M. Sc. (Applied Mathematics)

This thesis deals with different computational techniques related to some classes of nonlinear response regimes of engineering interest. The work is mainly divided into two parts. In the first part different numeric-analytic integration techniques for nonlinear oscillators are developed. In the second part, procedures for handling arbitrarily large rotations are addressed and a few novel developments are reported in the process. To begin the first part, we have proposed an explicit numeric-analytic technique, based on the Adomian decomposition method, for integrating strongly nonlinear oscillators. Numerical experiments suggest that this method, like most other numerical techniques, is versatile and can accurately solve strongly nonlinear and chaotic systems with relatively larger step-sizes. It is then demonstrated that the procedure may also be effectively employed for solving two-point boundary value problems with the help of a shooting algorithm. This has been followed up with the derivation and numerical exploration of variants of a recently developed numeric-analytic technique, the multi-step transversal linearization (MTrL), in the context of nonlinear oscillators of relevance in engineering dynamics. A considerable generalization and improvement over the original form of a MTrL strategy is achieved in this study. Finally, we have used the concept of MTrL method on the nonlinear variational (rate) equation corresponding to a nonlinear oscillator and thus derive another family of numeric-analytic techniques, presently referred to as the multi-step tangential linearization (MTnL). A comparison of relative errors through the MTrL and MTnL techniques consistently indicate a superior quality of approximation via the MTrL route. In the second part of the thesis, a scheme for numerical integration of rigid body rotation is proposed using only rudimentary tensor analysis. The equations of motion are rewritten in terms of rotation vectors lying in same tangent spaces, thereby facilitating vector space operations consistent with the underlying geometric structure of rotation. One of the most important findings of this part of the dissertation is that the existing constant-preserving algorithms are not necessarily accurate enough and may not be ideally applicable to cases wherein numerical accuracy is of primary importance. In contrast, the proposed rotation-algorithms, the higher order ones in particular, are significantly more accurate for conservative rotational systems for reasonably long time. Similar accuracy is expected for dissipative rotational systems as well. The operators relating rotation variables corresponding to different tangent spaces are also investigated and this should provide further insight into the understanding of rotation vector parametrization. A rotation update is next proposed in terms of rotation vectors. This update, employed along with interpolation of relative rotations, gives a strain-objective and path independent finite element implementation of a geometrically exact beam. The method has the computational advantage of requiring considerably less nodal variables due to the use of rotation vector parametrization. We have proposed a new isoparametric interpolation of nodal quaternions for computing the rotation field within an element. This should be a computationally efficient alternative to the interpolation of local rotations. It has been proved that the proposed interpolation of rotation leads to the objectivity of strain measures. Several numerical experiments are conducted to demonstrate the frame invariance, path-independence and other superior aspects of the present approach vis-`a-vis the existing methods based on the rotation vector parametrization. It is emphasized that, in order to develop an objective finite element formulation, the use of relative rotation is not mandatory and an interpolation of total rotation variables conforming with the rotation manifold should suffice.

The thesis deals with qualitative properties of solutions of singular initial value problems for ordinary differential and integrodifferential equations which occur in the theory of linear and nonlinear electrical circuits and the theory of therminionic currents. The research is concentrated especially on questions of existence and uniqueness of solutions, asymptotic estimates of solutions and modications of Adomian decomposition method for singular initial problems. Solution algoritms are derived for scalar differential equations of Lane-Emden type using Taylor series and modication of the Adomian decomposition method. For certain classes of nonlinear of integrodifferential equations asymptotic expansions of solutions are constructed in a neighbourhood of a singular point. By means of the combination of Wazewski's topological method and Schauder xed-point theorem there are proved asymptotic estimates of solutions in a region which is homeomorphic to a cone having vertex coinciding with the initial point. Using Banach xed-point theorem the uniqueness of a solution of the singular initial value problem is proved for systems of integrodifferential equations of Volterra and Fredholm type including implicit systems. Moreover, conditions of continuous dependence of a solution on a parameter are determined. Obtained results are presented in illustrative examples.