Nonlinear Partial Dierential Equations Research Articles

A new partial differential equation for image inpainting

A considerable interest in the inpainting problem have attracted many researchers in applied mathematics community. In fact in the last decade, nonlinear high order partial dierential equations have payed a central role in high quality inpainting developments. In this paper, we propose a technique for inpainting that combines an anisotropic diusion process with an edge-corner enhancing shock ltering. This technique makes use of a partial differential equation that is based on a nonlinear structure tensor which increases the accuracy and robustness of the coupled diusion and shock ltering. A methodology of partition and adjustment is used to estimate the contrast parameters that control the strength of the diffusivity functions. We focus on restoring large missing regions in grey scale images containing complex geometries parts. Our model is extended to a three dimensional case, where numerical experimentations were carried out on lling brain multiple sclerosis lesions in medical images. The efficiency and the competitiveness of the proposed algorithm is numerically compared to other approaches on both synthetic and real images.

Nonlinear Partial Dierential Equations Model Related to Oxidation Pond Treatment System: A Case Study of mPHO at Taman Timor Oxidation Pond, Johor Bahru

This study presents a mathematical model examining wastewater pollutant removalthrough an oxidation pond treatment system. This model was developed to describethe reaction between microbe-based product mPHO (comprising Phototrophic bac-teria (PSB)), dissolved oxygen (DO) and pollutant namely chemical oxygen demand(COD). It consists of coupled advection-diusion-reaction equations for the microor-ganism (PSB), DO and pollutant (COD) concentrations, respectively. The couplingof these equations occurred due to the reactions between PSB, DO and COD to pro-duce harmless compounds. Since the model is nonlinear partial dierential equations(PDEs), coupled, and dynamic, computational algorithm with a specic numericalmethod, which is implicit Crank-Nicolson method, was employed to simulate the dy-namical behaviour of the system. Furthermore, numerical results revealed that theproposed model demonstrated high accuracy when compared to the experimental data.Keywords Oxidation pond; nonlinear PDEs; PSB; implicit Crank-Nicolson.

KAJIAN TENTANG METODE PERSAMAAN RICCATI PROYEKTIF

Abstract. In this paper, we discuss the derivation and application of a projective Riccatiequation method in solving nonlinear partial dierential equations. We also study themathematical aspects of the method and its limitations in some particular cases.Kata Kunci: Nonlinear partial dierential equations, projective Riccati equation method,dominant balance principle

THEORETICAL AND NUMERICAL STUDIES OF TWO POINT BOUNDARY VALUE PROBLEMS USING TRIGONOMETRIC AND EXPONENTIAL CUBIC B-SPLINES

A collocation method with di erent types of B-spline functions such as trigonometric and exponential cubic B-splinesfunctions are proposed from many researchers for studying the numerical solutions of many physical nonlinear partialdi erential equations and ordinary di erential equations. In this paper, a collocation nite di erence scheme based ontrigonometric and exponential cubic B-splines functions are investigated and applied for the numerical solution of two-pointboundary value problems. The convergence of trigonometric and exponential cubic B-splines is proved using diagonallydominant matrices. The numerical examples show that our method is very e ective and the maximum absolute error isacceptable.

LED-Based Photometric Stereo: Modeling, Calibration and Numerical Solution

We conduct a thorough study of photomet-ric stereo under nearby point light source illumination, from modeling to numerical solution, through calibration. In the classical formulation of photometric stereo, the luminous uxes are assumed to be directional, which is very dicult to achieve in practice. Rather, we use light-emitting diodes (LEDs) to illuminate the scene to be reconstructed. Such point light sources are very convenient to use, yet they yield a more complex photo-metric stereo model which is arduous to solve. We rst derive in a physically sound manner this model, and show how to calibrate its parameters. Then, we discuss two state-of-the-art numerical solutions. The rst one alternatingly estimates the albedo and the normals, and then integrates the normals into a depth map. It is shown empirically to be independent from the ini-tialization, but convergence of this sequential approach is not established. The second one directly recovers the depth, by formulating photometric stereo as a system of nonlinear partial dierential equations (PDEs), which are linearized using image ratios. Although the sequential approach is avoided, initialization matters a lot and convergence is not established either. Therefore, we introduce a provably convergent alternating reweighted least-squares scheme for solving the original system of nonlinear PDEs. Finally, we extend this study to the case of RGB images.

SYMMETRY CLASSIFICATION OF NEWTONIAN INCOMPRESSIBLE FLUID'S EQUATIONS FLOW IN TURBULENT BOUNDARY LAYERS

Lie-group method is applicable to both linear and nonlinear partial dierential equations, which leads to nd new solutions for partial dierential equations. Lie symmetry group method is applied to study Newtonian incompressible uid’s equations ow in turbulent boundary layers. (Flow and heat transfer of an incompressible viscous uid over a stretching sheet appear in several manufacturing processes of industry such as the extrusion of polymers, the cooling of metallic plates, the aerodynamic extrusion of plastic sheets, etc. In the glass industry, blowing, oating or spinning of bres are processes, which involve the ow due to a stretching surface.) The symmetry group and its optimal system are given, and group invariant solutions associated to the symmetries are obtained [9, 10]. Finally the structure of the Lie algebra such as Levi decomposition, radical subalgebra, solvability and simplicity of symmetries is given.

Nonlinear traveling waves for the skeleton of the Madden–Julian oscillation

The Madden{Julian Oscillation (MJO) is the dominant component of intraseasonal (30{90 days) variability in the tropical atmosphere. Here, traveling wave solutions are presented for the MJO skeleton model of Majda and Stechmann. The model is a system of nonlinear partial dierential equations that describe the evolution of the tropical atmosphere on planetary (10,000{ 40,000 km) spatial scales. The nonlinear traveling waves come in four types, corresponding to the four types of linear wave solutions, one of which has the properties of the MJO. In the MJO traveling wave, the convective activity has a pulse-like shape, with a narrow region of enhanced convection and a wide region of suppressed convection. Furthermore, an amplitude-dependent dispersion relation is derived, and it shows that the nonlinear MJO has a lower frequency and slower propagation speed than the linear MJO. By taking the small-amplitude limit, an analytic formula is also derived for the dispersion relation of linear waves. To derive all of these results, a key aspect is the model's conservation of energy, which holds even in the presence of forcing. In the limit of weak forcing, it is shown that the nonlinear traveling waves have a simple sech-squared waveform.

Application of Homotopy Perturbation Transform Method to Resolve Lane - Emden and Emden - Fowler Equations

In this study, we will apply homotopy perturbation transform method (HPTM) to solve Lane-Emden and Emden-Fowler equations and we will also make a comparison between the result obtained by this method and the exact solutions of the same equations presented in other research. The results showed that HPTM has a high eciency and eectiveness in solving these equations and can be applied to other nonlinear partial dierential equations to obtain approximate or exat solutions

A Parabolic Transform and Some Stochastic Ill-Posed Problems

It is well known that the Cauchy problem for elliptic partial dierential equations is ill-posed. The question, which arises, how a priori knowledge about solutions can bring about stability? A parabolic transform is dened to discuss the stability of the Cauchy problem for some stochastic partial dierential equations under a priori knowledge about solutions. With the help of the parabolic transform, existence results are established for general linear and nonlinear stochastic partial dierential equations, without any restrictions on the characteristic forms. Many physical and engineering problems in areas like seismology, geophysics and biology require the solutions of ill-posed problems. The Cauchy problem for general stochastic dierential equations has many dierent important applications with amazing range.

Analytic continuation of solutions of some nonlinear convolution partial differential equations

The paper considers a problem of analytic continuation of solutions of some nonlinear convolution partial dierential equations which naturally appear in the summa- bility theory of formal solutions of nonlinear partial dierential equations. Under a suitable assumption it is proved that any local holomorphic solution has an analytic extension to a certain sector and its extension has exponential growth when the variable goes to infinity in the sector.

A SIMPLE NASH-MOSER IMPLICIT FUNCTION THEOREM IN WEIGHTED BANACH SPACES

We prove a simplified version of the im- plicit function theorem in weighted Banach spaces. We relax the conditions so that the linearized equation has an approximate in- verse in dierent weighted Banach spaces in each recurrence step. During the last several decades, Nash-Moser implicit function theo- rem helped to resolve several dicult problems of solvability for non- linear problems, especially, nonlinear partial dierential equations (6, 11). Usually nonlinear partial dierential equations (or nonlinear problems in general) can be transformed into solving the problem : `(u) = 0; whereinvolves the variables x, the unknown function u(x) and its derivatives up to the order m. To prove implicit function theorem in infinite dimensional spaces (as spaces of functions usually are), we first linearize the equation, and then solve the linear equation so that we get the recursive solutions with ap- propriate recurrence estimates. The simplest one is known as Picard's iterative scheme. However wheninvolves the derivatives of u up to or- der m, the Picard's scheme can not be convergent unless the linearized equation gets m derivatives (as does elliptic equations). To overcome this diculty, Nash (11) and Moser (10) proposed another scheme involv- ing smoothing operators so that the solution of the linearized equation

Sensitivity Analysis of a Nonlinear Obstacle Plate Problem

PRAXIS/PCEX/C/MAT/38/96 \Variational Models and Optimization, and Sapiens Proj99, No. 34471 \Nonlinear Partial Di erential Equations and Interfaces Problems of the Ministry of Science and Technology of Portugal

Gauge symmetry and the generalization of Hirota's bilinear method

One of the most powerful methods for finding and solving integrable nonlinear partial dierential equations is Hirota’s bilinear method. The idea behind it is to make first a nonlinear change in the dependent variables after which multisoliton solutions of integrable systems can be expressed as polynomials of exponentials e i where the 0s are linear in the independent variables. Among all quadratic expressions homogeneous in the derivatives, Hirota’s bilinear form can be isolated by a gauge symmetry: it is the only one that is invariant under f ! e f where is linear in the variables. This suggest a generalization to multilinear equations using the same gauge symmetry. The set of gauge invariant multilinear dierential equations can then be studied and

The Singular Manifold Method: Darboux Transformations and Nonclassical Symmetries

We present in this paper the singular manifold method (SMM) derived from Painleve analysis, as a helpful tool to obtain much of the characteristic features of nonlinear partial dierential equations. As is well known, it provides in an algorithmic way the Lax pair and the Backlund transformation for the PDE under scrutiny. Moreover, the use of singular manifold equations under homographic invariance consideration leads us to point out the connection between the SMM and so-called nonclassical symmetries as well as those obtained from direct methods. It is illustrated here by means of some examples. We introduce at the same time a new procedure that is able to determine the Darboux transformations. In this way, we obtain as a bonus the one and two soliton solutions at the same step of the iterative process to evaluate solutions.