In this paper, we present a novel technique based on the Legendre wavelets decomposition. The properties of Legendre wavelets are used to reduces the **PDEs** problem into the solution of ODEs system. To illustrate our results, two examples are studied using a special software package which implements the proposed algo- rithms.

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A number of methods have been developed to deal with the numerical solution of **PDEs**. These fall into two broad categories: the finite-difference methods and the finite-element methods. Roughly speaking, both transform a PDE problem to the problem of solving a system of coupled algebraic equations. In finite-difference methods, the domain of the in- dependent variables is approximated by a discrete set of points called a grid, and the dependent variables are defined only at these points. Any derivative then automatically acquires the meaning of a certain kind of difference between dependent variable values at the grid points. This identification helps us transform the PDE problem to a system of coupled algebraic equations. In the finite-element method, the dependent variable is approximated by an interpolation polynomial. The domain is subdivided into a set of non- overlapping regions that completely cover it; each such region is called a finite element. The interpolation polynomial is determined by a set of coefficients in each element; the small size of the elements generally ensures that a low-degree polynomial is sufficient. A weighted residual method is then used to set up a system of algebraic equations for these coefficients. The numerical problem in both methods therefore is one of solving a system of algebraic equations.

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The paper [43] is written for the graduate, or even the professional research mathematician; this thesis aims to explain the same material to a late undergraduate or beginning graduate audience. However, the reader should have undergraduate or better experience with Banach space theory, measure theory and Lebesgue integration, and at least some familiarity with complex analysis and probability. An understanding of **PDEs** is recommended to motivate the results, but the thesis uses functional analytic techniques and no PDE familiarity is required to follow any discussions or proofs. Harmonic analysis techniques do come into play in the study of Maximal L p -regularity, but these considerations are beyond this thesis’ scope and are not

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In this paper, a local meshless method (LMM) based on radial basis functions (RBFs) is utilized for the numerical solution of various types of **PDEs**, due to the flexibility with respect to geometry and high order of convergence rate. In case of global meshless methods, the two major deficiencies are the computational cost and the optimum value of shape parameter. Therefore, research is currently focus towards localized radial basis function approximations, as the local meshless method is proposed here. The local meshless procedures is used for spatial discretization whereas for temporal discretization different time integrators are employed. The proposed local meshless method is testified in terms of efficiency, accuracy and ease of implementation using regular and irregular domains.

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• Another approach is to numerically solve bulk equations in one space dimension higher. This may be a natural approach when the surface is computed implicitly using phase field or level set methods or when one wishes to use bulk finite element codes. There are a number of variants: – One idea is to solve the surface partial differential equations on all level sets of a prescribed function yielding degenerate bulk **PDEs**. See [9, 43, 42, 27] for stationary surfaces. In the context of con- servation laws on moving surfaces this is inherently an Eulerian approach see [1, 63] where level set approximations to surface quantities such as the mean curvature and normal velocity were required. On the other hand an elegant formulation avoiding the need to do this was provided in [28] using an implicit surface ver- sion of the Leibniz formula. The idea is then to exploit the implicit formulation and rather than a surface triangulation use a bulk tri- angulation which is independent of the surface. In practice it may be useful to solve in a narrow band and use unfitted bulk finite elements, [3]. For surface elliptic equations, [15] gave a discretisa- tion error analysis for a narrow band level set method using the unfitted finite element method.

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The computational complexity of the algorithm is as fol- lows. Since we are computing the **PDEs** in Eqs. (7) and (10) over a narrow-band around the zero level set of the sur- face in R 3 (usually radius of the band is chosen as 5), the general complexity is O(N 2 ). For the computation of the means inside and outside the image volume for Eq. (7), the complexity is O(N 2 ) except at initialization it is O(N 3 ) (going through all the image volume). For the computation of the means inside and outside the second image volume for Eq. (10) or (12), the second band (surface) is initialized after the first band (surface) is updated every time, therefore with the current implementation the complexity is O(N 3 ), but it is possible to compute it at O(N 2 ).

system of ODEs are solved analytically. Any method can be used to discretised the independent variables. This includes Fourier Transform or the finite differ ence method. The technique being used in this thesis was to replace all the partial derivatives with the central finite difference approximation that gives a system of ODEs. Although this formulation may differ from other approaches, it is clearly advocated by [Liu et al., 2004; Lord et al., 2014; Trefethen, 2000] as an alternative approach, as the fundamental principles are the same. The method of lines which is used in the thesis involves solving the elliptic **PDEs** over a rectangular domain. The domain is defined by mesh cutting planes yielding vertices on a regular grid that define the top and bottom boundaries of the domain. All vertices on the top boundary can be paired to their corresponding vertices on the opposite bottom boundary. The left and right boundaries are defined by interpolating between the first top and first bottom vertices and last top and last bottom vertices using the finite difference method. The number of interpolated vertices is user defined. All interior vertices to the rectangular domain are initialised to zero and are interpo lated by iteratively solving Laplace’s equation over the domain. Therefore, we approximate Laplace’s equation at each grid point, and the resulting equations are solved by iteration through implementing the Matlab function 'gmprLaplace .m '. Further description is given in Section 6.2.3.

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Cyclic nucleotide **PDEs** carry out essential roles in signal transduction by modulating cAMP and cGMP levels and have been recognized as potential targets of several blad- der diseases such as overactive bladder. The presence and activities of PDE1–5 have been tested and con- firmed in the rodent, rabbit, guinea pig, and human urinary bladder by pharmacological, biochemical, immu- nohistochemical and molecular methods [12–16]. By using qPCR technique, Lakics et al. have shown that sev- eral PDE isoform of PDE7–10 were expressed in the hu- man urinary bladder [16]. Moreover, the expression of PDE5 in different cell types of urinary bladder such as the urothelium, suburothelial interstitial cells and blad- der blood smooth muscle cells has been determined in the guinea pig [17]. However, the molecular levels of the whole **PDEs** family in the rodent urinary bladder are still unknown. Here, we established the mRNA expression levels of 18 different PDE isoforms in rat urinary bladder by RT-PCR. Our results indicated that PDE5 is one of the major isoform expressed in rat urinary bladder, Table 2 The expression levels of PDE isoforms in adult rat urinary bladder

In conclusion, we presented efficient and simple image segmentations based on ideas from the Eikonal and diffusion **PDEs**, by computing the distance functions for the exterior and interior regions, and determining the final segmentation labels by a competition criterion between the distance functions for reaching a given point. Each method has its pros and cons, according to the image characteristics, but our exper- iments demonstrated that among the presented methods, the combined fast marching method achieved a better speed vs. accuracy ratio, hence the best utility when com- pared to the other three methods.

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For decades now, integrable partial differential equations (**PDEs**), and more generally integrable systems, have fuelled research and important discoveries in Mathematics and Physics, and still do. Comparatively more recently, graphs and dynamical systems on (quantum) graphs have emerged as a successful framework to model a large variety of (complex) systems. It is therefore not surprising to see a fast growing interest in developing a theory of integrable systems on graphs, which would combine the power of integrable systems with the flexibility of graphs to model more realistic situations. The review [1], for instance, gives a flavour and references for this fast growing area in the context of nonlinear Schr¨odinger (NLS) equations (not restricted to integrable cases). Originally, integrable **PDEs** were treated as initial value problems for functions of one space variable x ∈ R and one time variable t ≥ 0. The invention of the inverse scattering method (ISM) [2] and its refinements [3, 4] through the systematic use of a Lax pair [5] represents a cornerstone of modern integrable **PDEs**. The first departure from this setup to solve an initial-boundary value (IBV) problem for an integrable PDE on the half-line [6, 7] or a finite interval [8] can be viewed in retrospect as the beginning of the study of integrable **PDEs** on metric graphs. Indeed, a half-line is nothing but a half-infinite edge attached to a vertex and a finite interval is a finite edge connecting two vertices. The next big step in this natural evolution was the study of integrable **PDEs** on the line with a defect/impurity at a fixed site (or possibly several such defects). The vast literature on this problem 1 [9, 10, 11, 12, 13, 14, 15, 16, 17, 18] shows both its interest and its difficulty. To date however, despite some impressive results on the behaviour of certain solutions [19, 20], the general problem of formulating an ISM for a problem with defects is still open.

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For decades now, integrable partial differential equations (**PDEs**), and more generally integrable systems, have fuelled research and important discoveries in Mathematics and Physics, and still do. Comparatively more recently, graphs and dynamical systems on (quantum) graphs have emerged as a successful framework to model a large variety of (complex) systems. It is therefore not surprising to see a fast growing interest in developing a theory of integrable systems on graphs, which would combine the power of integrable systems with the flexibility of graphs to model more realistic situations. The review [1] for instance gives a flavour and references for this fast growing area in the context of nonlinear Schr¨ odinger (NLS) equations (not restricted to integrable cases). Originally, integrable **PDEs** were treated as initial value problems for functions of one space variable x ∈ R and one time variable t ≥ 0. The invention of the inverse scattering method (ISM) [2] and its refinements [3, 4] through the systematic use of a Lax pair [5] represents a cornerstone of modern integrable **PDEs**. The first departure from this setup to solve an initial-boundary value (IBV) problem for an integrable PDE on the half-line [6, 7] or a finite interval [8] can be viewed in retrospect as the beginning of the study of integrable **PDEs** on metric graphs. Indeed, a half-line is nothing but a half-infinite edge attached to a vertex and a finite interval is a finite edge connecting two vertices. The next big step in this natural evolution was the study of integrable **PDEs** on the line with a defect/impurity at a fixed site (or possibly several such defects). The vast literature on this problem 1 [9, 10, 11, 12, 13, 14, 15, 16, 17, 18] shows both its interest and its difficulty. To date however, despite some impressive results on the behaviour of certain solutions [19, 20], the general problem of formulating an ISM for a problem with defects is still open.

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This thesis concerns itself with an analytical and numerical study of a family of evolutionary partial differential equations (**PDEs**) which supports peakon solutions for special values of a given bifurcation parameter. Here, the bifurcation parameter describes the balance between convection and stretching for small viscosity in the dynamics of one dimensional (1D) nonlinear waves in fluids.The first portion of this thesis is to provide global existence and uniqueness results for the considered family of evolutionary **PDEs** by establishing convergence results for the particle method applied to these equations. This particular class of **PDEs** is a collection of strongly nonlinear equations which yield traveling wave solutions and can be used to model a variety of flows in fluid dynamics. We apply a particle method to the studied evolutionary equations and provide a new self-contained method for proving its convergence. The latter is accomplished by using the concept of space-time bounded variation and the associated compactness properties. From this result, we prove the existence of a unique global weak solution in some special cases and obtain stronger regularity properties of the solution than previously established. The second portion of this thesis is dedicated to studying the dynamics of the in- teraction among a special class of solutions of the one-dimensional Camassa-Holm (CH) equation which are a particular example of such a PDE which supports peakon solutions. The equation yields soliton solutions whose identity is preserved through nonlinear in- teractions. These solutions are characterized by a discontinuity at the peak in the wave shape and are thus called peakon solutions. We apply a particle method to the CH equa- tion and show that the nonlinear interaction among the peakon solutions resembles an elastic collision, i.e., the total energy and momentum of the system before the peakon interaction is equal to the total energy and momentum of the system after the collision. From this result, we provide several numerical illustrations which supports the analytical study, as well as showcase the merits of using a particle method to simulate solutions to the CH equation under a wide class of initial data.

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3. Theory of elasticity and Colombeau’s algebra. In this section, we consider a simpliﬁed model of elasticity to apply new generalized functions for investigating jump conditions of nonlinear **PDEs** arising in this theory. Jump conditions are important in elasticity since these conditions contain information about the behavior of the system on the boundary and wavefront. On the other hand, a part of the structural properties about the medium are contained in these jump conditions. In the system of elasticity, Hooke’s law in terms of the stress σ can be expressed as (d/dt)σ = k 2 u

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We study Poisson and Lie-Poisson structures on the diffeomorphism groups with a smooth metric spray in connection with dynamics of nonlinear **PDEs**. In particular, we provide a precise analytic sense in which the time t map for the Euler equations of an ideal fluid in a region of R n (or on a smooth compact n-manifold with a bound- ary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphisms. The key difficulty in finding a suitable context for that arises from the fact that the integral curves of Euler equations are not dif- ferentiable on the Lie algebra of divergence free vector fields of Sobolev class H s . We overcome this obstacle by utilizing the smoothness that one has in Lagrangian repre- sentation and carefully performing a non-smooth Lie-Poisson reduction procedure on the appropriate functional classes.

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Over the last few decades, there has been much progress towards making sense of solutions to stochastic **PDEs**, where the forcing term may be a highly irregular Gaussian signal taking values in spaces of rather irregular distributions, see for example [3, 7] for introductory texts on the subject. It is therefore natural to ask whether asymptotic results for **PDEs** like (1.4) can be extended to the case where f is a random, distribution- valued process. To give an idea of the type of results obtained in this article, let ξ be space-time white noise, which is the distribution-valued Gaussian process formally satisfying E ξ(s, x)ξ(t, y) = δ(s − t)δ(x − y) . For fixed ε > 0 , one can easily show that

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The starting point was the use of surface ﬁnite elements to compute solu- tions to the Poisson problem for the Laplace–Beltrami operator on a curved surface proposed and analysed in Dziuk (1988). Here an important con- cept is the use of triangulated surfaces on which ﬁnite element spaces are constructed and then used in variational formulations of surface **PDEs** us- ing surface gradients. This approach was extended by Dziuk and Elliott (2007b) to parabolic (including nonlinear and higher-order) equations on stationary surfaces. The evolving surface ﬁnite element method (ESFEM) was introduced by Dziuk and Elliott (2007a) in order to treat conservation laws on moving surfaces. The key idea is to use the Leibniz (or transport) formula for the time derivative of integrals over moving surfaces in order to derive weak and variational formulations. An interesting upshot is that the velocity and mean curvature of the surface which appear in certain for- mulations of the partial diﬀerential equation do not appear explicitly in variational formulations. This gives a tremendous advantage to numerical methods that exploit this, such as those of Dziuk and Elliott (2007a, 2010). Further numerical analysis of surface ﬁnite element methods may be found in Dziuk and Elliott (2012, 2013) and Dziuk, Lubich and Mansour (2012). Applications to complex physical and biological models may be found in Eilks and Elliott (2008), Elliott and Stinner (2010), Barreira, Elliott and Madzvamuse (2011) and Elliott, Stinner and Venkataraman (2012).

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In this paper, we applied relatively new fractional complex transform (FCT) to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (**PDEs**) and Variational Iteration Method (VIM) is to find approximate solution of time- fractional Fornberg-Whitham and time-fractional Wu-Zhang equations. The results so obtained are re-stated by making use of inverse transformation which yields it in terms of original variables. It is observed that the proposed algorithm is highly efficient and appropriate for fractional **PDEs** arising in mathematical physics and hence can be extended to other problems of diversified nonlinear nature. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the proposed algorithm.

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The contribution of this paper is an algorithm for joint segmentation and registration in infinite dimensions through coupling of two **PDEs** for surface and deformation field evolutions. Note that the desired coupling comes from estimating the common surface and its non-rigid deforma- tions onto a target image. The solutions of these two **PDEs** both decrease the total energy of the surface, therefore aid each other in finding a locally optimal solution to the prob- lem. The success of the algorithm in its current form de- pends on how well the region descriptor f represents the regions in the images, however, it can utilize more compli- cated statistical descriptors and information-theoretic mea- sures, a direction to be explored. For target regions with large image clutter, inclusion of shape priors may be neces- sary. In that case, one can incorporate into this framework as well a mean shape from a training phase to penalize the surface deviations from the desired shape model. However, the framework we propose is general in the sense that inclu-

We begin by taking a look at the modified output-least squares (MOLS) functional that emerged as an alternative to the generally non-convex output least-squares (OLS) functional. The MOLS has the desirable property of being a convex functional, which was shown in [1]. However, the convexity of the MOLS has only been established for parameters appearing linearly in the **PDEs**. The primary objective of this chapter is to introduce and analyze a variant of the MOLS for the inverse problem of identifying parameters that appear nonlinearly in general variational problems. We are interested in understanding what geometric properties of the original MOLS can be retained for the nonlinear case. Besides giving an existence result for the optimization formulation of the inverse problem, we give a thorough derivation of the first-order and second-order derivative formulas for the new objective functional. The derivative formulae suggest that the convexity of the MOLS cannot be retained for the parameters appearing nonlinearly without imposing additional assumptions on the data involved.

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Partial Differential Equations (**PDEs**) on evolving or stationary surfaces are ubiqui- tous in the mathematical modelling of real world phenomena. Problems in biology, engineering and image analysis all benefit from the rich mathematical literature on such objects. For example, Neilson et al. [2011] studies chemotaxis in cells and uses a non-linear PDE system to describe the evolution of the quantities on an evolv- ing curve. Moreover, the evolution of the curve is strongly coupled to dynamics of the quantities. Surface dissolution (which has applications in the mining of gold, for example), is modelled by an evolving surface whose evolution is coupled to a non-linear surface PDE (Eilks and Elliott [2008]). In image analysis, motion by mean curvature has been used to describe the graph of a surface which represents an image, to recover the image (Faugeras and Keriven [2002]).

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