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Subject: search.filters.subject.Differential equations, Partial.

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Now showing 1 - 10 of 12
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    Lie group approach to Cauchy's problems : solution of an initial value problem for the Black-Scholes model
    (1997) Wafo, Soh, Celestin
    A Lie group assisted method is used to solve explicitly an arbitrary initial value problem for the Black-Scholes equation, This equation plays a crucial role in the mathematics of finance. It was first solved by its inventors for a special initial data. Our solution generalises the well-known Black-Scholes formula.
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    Lie group approach to Cauchy's problems : solution of an initial value problem for the Black-Scholes model
    (1997) Soh, Celestin Wafo
    A Lie group assisted method is used to solve explicitly an arbitrary initial value problem for the Black-Scholes equation, This equation plays a crucial role in the mathematics of finance. It was first solved by its inventors for a special initial data. Our solution generalises the well-known Black-Scholes formula.
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    Equivalence and symmetry groups of a nonlinear equation in plasma physics
    (2016-07-14) Bashe, Mantombi Beryl
    In this work we give a brief overview of the existing group classification methods of partial differential equations by means of examples. On top of these methods we introduce another new method which classify according to low-dimensional Lie elgebras, One can ask: What is the aim of introducing a new method whilst there are existing methods? This question is answered in the following paragraph. Firstly we classify our system of non-linear partial differential equations using the preliminary group classification method (one of the existing methods). The results are not different from what; Euler, Steeb and Mulsor have obtained in 1991 and 1992. That is, this method does not yield new information. This new method which classifies according to low-dimensional Lie algebras is used to classify a general system of equations from plasma physics. Finally, using this method we completely classify our system for four-dimensionnl algebras. For a partial differential equation to be completely classified using this method, it must admit a low-dimensional Lie algebra.
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    Smoothness conditions and symmetries of partial differential equations
    (2016-05-10) Mamba, Siphamandla
    We obtain a solution of the Black-Scholes equation with a non-smooth bound- ary condition using symmetry methods. The Black-Scholes equation along with its boundary condition are rst transformed into the one dimensional heat equation and an initial condition respectively. We then nd an appro- priate general symmetry generator of the heat equation using symmetries of the heat equation and the fundamental solution of the heat equation. The method we use to nd the symmetry generator is such that the boundary condition is left invariant and yet the symmetry can still be used to solve the heat equation. We then use the help of Mathematica to nd the solution to the heat equation. Then the solution is then transformed backwards to a solution of the Black-Scholes equation using the same change of variables that were used for the forward transformations. The solution is then nally checked if it satis es the boundary condition of the Black-Scholes equation.
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    Double reduction of partial differential equations with applications to laminar jets and wakes
    (2016) Kokela, Lady Nomvula
    Invariant solutions for two-dimensional free and wall jets are derived by consid- ering the Lie point symmetry associated with the appropriate conserved vectors of Prandtl's boundary layer equations for the jets. For the two-dimensional jets we also consider the comparison, advantages and disadvantages between the standard method that uses a linear combination of all the Lie point symme- tries of Prandtl's boundary layer equations to generate the invariant solution with the new method explored in this paper which uses the Lie point sym- metry associated with a conserved vector to generate the invariant solution. Invariant solutions for two-dimensional classical and self-propelled wakes are also derived by considering the Lie point symmetry associated with the appro- priate conserved vectors of Prandtl's boundary layer equations for the wakes. We also consider and discuss the standard method that uses a linear combi- nation of all the Lie point symmetry of Prandtl's boundary layer equations to generate the invariant solutions for the classical and self-propelled wakes.
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    Invariances, conservation laws and conserved quantities of the two-dimensional nonlinear Schrodinger-type equation
    (2014) Lepule, Seipati
    Symmetries and conservation laws of partial di erential equations (pdes) have been instrumental in giving new approaches for reducing pdes. In this dissertation, we study the symmetries and conservation laws of the two-dimensional Schr odingertype equation and the Benney-Luke equation, we use these quantities in the Double Reduction method which is used as a way to reduce the equations into a workable pdes or even an ordinary di erential equations. The symmetries, conservation laws and multipliers will be determined though di erent approaches. Some of the reductions of the Schr odinger equation produced some famous di erential equations that have been dealt with in detail in many texts.
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    Group theoretical and compatibility approaches to some nonlinear PDEs arising in the study of non-Newtonian fluid mechanics
    (2015-05-06) Aziz, Taha
    This thesis is primarily concerned with the analysis of some nonlinear problems arising in the study of non-Newtonian fluid mechanics by employing group theoretic and compatibility approaches. It is well known now that many manufacturing processes in industry involve non-Newtonian fluids. Examples of such fluids include polymer solutions and melts, paints, blood, ketchup, pharmaceuticals and many others. The mathematical and physical behaviour of non-Newtonian fluids is intermediate between that of purely viscous fluid and that of a perfectly elastic solid. These fluids cannot be described by the classical Navier–Stokes theory. Striking manifestations of non-Newtonian fluids have been observed experimentally such as the Weissenberg or rod-climbing effect, extrudate swell or vortex growth in a contraction flow. Due to diverse physical structure of non-Newtonian fluids, many constitutive equations have been developed mainly under the classification of differential type, rate type and integral type. Amongst the many non-Newtonian fluid models, the fluids of differential type have received much attention in order to explain features such as normal stress effects, rod climbing, shear thinning and shear thickening. Most physical phenomena dealing with the study of non-Newtonian fluids are modelled in the form of nonlinear partial differential equations (PDEs). It is easier to solve a linear problem due to its extensive study as well due to
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    Sparse array representations and some selected array operations on GPUs
    (2014-09-01) Wang, Hairong
    A multi-dimensional data model provides a good conceptual view of the data in data warehousing and On-Line Analytical Processing (OLAP). A typical representation of such a data model is as a multi-dimensional array which is well suited when the array is dense. If the array is sparse, i.e., has a few number of non-zero elements relative to the product of the cardinalities of the dimensions, using a multi-dimensional array to represent the data set requires extremely large memory space while the actual data elements occupy a relatively small fraction of the space. Existing storage schemes for Multi-Dimensional Sparse Arrays (MDSAs) of higher dimensions k (k > 2), focus on optimizing the storage utilization, and offer little flexibility in data access efficiency. Most efficient storage schemes for sparse arrays are limited to matrices that are arrays in 2 dimensions. In this dissertation, we introduce four storage schemes for MDSAs that handle the sparsity of the array with two primary goals; reducing the storage overhead and maintaining efficient data element access. These schemes, including a well known method referred to as the Bit Encoded Sparse Storage (BESS), were evaluated and compared on four basic array operations, namely construction of a scheme, large scale random element access, sub-array retrieval and multi-dimensional aggregation. The four storage schemes being proposed, together with the evaluation results are: i.) The extended compressed row storage (xCRS) which extends CRS method for sparse matrix storage to sparse arrays of higher dimensions and achieves the best data element access efficiency among the methods compared; ii.) The bit encoded xCRS (BxCRS) which optimizes the storage utilization of xCRS by applying data compression methods with run length encoding, while maintaining its data access efficiency; iii.) A hybrid approach (Hybrid) that provides the best control of the balance between the storage utilization and data manipulation efficiency by combining xCRS and BESS. iv.) The PATRICIA trie compressed storage (PTCS) which uses PATRICIA trie to store the valid non-zero array elements. PTCS supports efficient data access, and has a unique property of supporting update operations conveniently. v.) BESS performs the best for the multi-dimensional aggregation, closely followed by the other schemes. We also addressed the problem of accelerating some selected array operations using General Purpose Computing on Graphics Processing Unit (GPGPU). The experimental results showed different levels of speed up, ranging from 2 to over 20 times, on large scale random element access and sub-array retrieval. In particular, we utilized GPUs on the computation of the cube operator, a special case of multi-dimensional aggregation, using BESS. This resulted in a 5 to 8 times of speed up compared with our CPU only implementation. The main contributions of this dissertation include the developments, implementations and evaluations of four efficient schemes to store multi-dimensional sparse arrays, as well as utilizing massive parallelism of GPUs for some data warehousing operations.
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    On the application of partial differential equations and fractional partial differential equations to images and their methods of solution
    (2014-08-11) Jacobs, Byron
    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.
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    Pricing of double barrier options from a symmetry group approach
    (2014-07-02) Sidogi, Thendo
    In this research report we explore some applications of symmetry methods for boundary value problems in the pricing of barrier options. Various nancial instruments satisfy the Black-Scholes partial di erential equation (pde) but with di erent domain, maturity date and boundary conditions. We nd Lie symmetries that leave the Black-Scholes (pde) invariant and will guarantee that the relevant solutions satisfy the boundary conditions. Using these sym- metries, we can thus generate group-invariant solutions to the boundary value problem.