Partial Differential Equations

To model the physical problems, the partial differential equations (PDEs) are the common method. PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, gravitation and quantum mechanics, etc. In this chapter, we will discuss about different types of the partial differential equations, their classifications and the classical and weak solutions, etc.

The Physical system contains arbitrary constants or arbitrary functions or both. Equations which contain one or more partial derivatives are called Partial Differential Equations. Therefore, there must be atleast two independent variables and one dependent variable. Let us take to be two independent variables and to be dependent variable.

Abstract The primary objective of this research is to investigate various optimization problems connected with partial differential equations (PDE). In chapter 2, we utilize the tool of tangent cones from convex analysis to prove the exis- tence and uniqueness of a minimization problem. Since the admissible set considered in chapter 2 is a suitable convex set in L ∞ (D), we can make use of tangent cones to derive the optimality condition for the problem. How- ever, if we let the admissible set to be a rearrangement class generated by a general function (not a characteristic function), the method of tangent cones may not be applied. The central part of this research is Chapter 3, and it is conducted based on the foundation work mainly clarified by Ge- offrey R. Burton with his collaborators near 90s, see [7, 8, 9, 10]. Usually, we consider a rearrangement class (a set comprising all rearrangements of a prescribed function) and then optimize some energy functional related to partial differential equations on this class or part of it. So, we call it rearrangement optimization problem (ROP). In recent years this area of re- search has become increasingly popular amongst mathematicians for several reasons. One reason is that many physical phenomena can be naturally formulated as ROPs. Another reason is that ROPs have natural links with other branches of mathematics such as geometry, free boundary problems, convex analysis, differential equations, and more. Lastly, such optimization problems also offer very challenging questions that are fascinating for re- searchers, see for example [2]. More specifically, Chapter 2 and Chapter 3 are prepared based on four papers [24, 40, 41, 42], mainly in collaboration with Behrouz Emamizadeh. Chapter 4 is inspired by [5]. In [5], the exis- tence and uniqueness of solutions of various PDEs involving Radon measures are presented. In order to establish a connection between rearrangements and PDEs involving Radon measures, the author try to investigate a way to extend the notion of rearrangement of functions to rearrangement of Radon measures in Chapter 4.

3.1.2 Outline of chapter The chapter is laid out as follows. In section two, we will derive a Cahn-Hilliard equation on an evolving surface using a local conservation law. We introduce the notation for partial differential equations on evolving surfaces following Deckelnick et al. (2005) and state any assumptions on the smoothness of the surfaces and its evolution we require in later chapters. The third section introduces a finite element discretisation of the continuous equations. We describe the process of triangulat- ing an evolving surface and how we formulate the space discrete-time continuous problem as a system of ordinary differential equations. This section is completed by showing some domain perturbation results relating geometric quantities on the dis- crete and smooth surfaces. Well-posedness of the continuous equations is addressed in the fourth section. An existence result is achieved by showing convergence, along a subsequence, of the discrete solutions as the mesh size tends to zero. In section five, we analyse the errors introduced by our finite element scheme and go on to show an optimal order error estimate. Some numerical experiments are shown in the sixth section backing up the analytical results.

Not always is the function analytically known (but we are usually able to compute the function numerically) The material presented here forms the basis of the finite-difference technique that is commonly used to solve ordinary and partial differential equations. The following slides show

Emmanuel Grenier, Marc Hoffmann, Tony Leli` evre, Violaine Louvet, Cl´ ementine Prieur, Nabil Rachdi, Paul Vigneaux To cite this version: Emmanuel Grenier, Marc Hoffmann, Tony Leli` evre, Violaine Louvet, Cl´ ementine Prieur, et al.. Statistical Inference for Partial Differential Equations. SMAI 2013 - 6e Biennale Fran¸caise des Math´ ematiques Appliqu´ ees et Industrielles, May 2013, Seignosse, France. EDP Science, ESAM: ProcS, 45, pp.178-188, 2014, Congr` es SMAI 2013. <10.1051/proc/201445018>. <hal- 01102782>

In Chapter 2 we establish basic properties of the the eigenvalues and eigenfunc­ tions for the Laplacian on p.c.f. fractals using the Sobolev-type inequality of Chapter 1 . T he techniques are standard but we include the results for com pleteness. Chapter 3 goes to construct the fundamental solutions such as heat kernels, G reen’s functions and wave propagators on fractals. T he results of Chapter 2 are extensively exploited. These fundam ental solutions are the corner stones for studying non-linear partial differential equations on fractals. T hey present a striking contrast to their counterparts on classical domains. T he reason for this is that the spectral dim ension is less than 2 in the fractal case.

network, optics and signal processing, electrochemistry, continuum mechanics and so on. However, fractional calculus is three centuries old as the conventional calculus [1].The most recent works on the subject of fractional calculus is the book of Podlubny [2]. The importance of obtaining the exact and approximate solutions offractional linear or nonlinear differential equations is still significant problem that need snew methods to discover the exact and approximate solutions. But these nonlineardifferential equations are difficult to get their exact solutions so numerical methods havebeen used to handle these equations, a wide class of analytical methods have been proposed, suchas Laplace transform method [2,3], differential transform method[4-6], Adomian’s decomposition method [7-11], variational iteration method [12-14], homotopy perturbation method [15-16], homotopy perturbation transform method [17]. Another analytical approach that can be applied to solve linear or nonlinear equations ishomotopy analysis method [18- 20]. A systematic and clear exposition on HAM is given in [19]. The objective of the paper is to apply the Laplace homotopy analysis method [21], to provide analytic approximate solutions to inhomogeneous fractional partial differential equations.

Why do we study partial differential equations (PDEs) and in particular analytic solutions? We are interested in PDEs because most of mathematical physics is described by such equa- tions. For example, fluids dynamics (and more generally continuous media dynamics), elec- tromagnetic theory, quantum mechanics, traffic flow. Typically, a given PDE will only be accessible to numerical solution (with one obvious exception — exam questions!) and ana- lytic solutions in a practical or research scenario are often impossible. However, it is vital to understand the general theory in order to conduct a sensible investigation. For example, we may need to understand what type of PDE we have to ensure the numerical solution is valid. Indeed, certain types of equations need appropriate boundary conditions; without a knowledge of the general theory it is possible that the problem may be ill-posed of that the method is solution is erroneous.

• Most dynamic effects and physical processes can be described by partial differential equations PDEs • We give a short overview of. a large research field in mathematics and physics[r]

Copyright © 2016 ISSR Journals. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A BSTRACT : The mathematical notion of nonlinear partial derivatives equations are registered in full in Mathematical Analysis. This concept has several applications in other disciplines such as physics, economics, demography, chemistry, differential geometry and infinitesimal, etc. Presented with diversified forms, this notion remains essential in the progress of all research in pure and applied mathematics. It borrows concepts of topology and functional analysis for a better understanding. The question is at what level mathematics are for research and our contribution in this article. For this, we present the theory and develop some methods of solving equations to nonlinear partial differential equations.

SPDEs are partial differential equations for which the input data (i.e. forcing terms, diffu- sion/advection/reaction coefficients, domain) are uncertain. This uncertainty could arise [r]

Keywords : existence and uniqueness, parabolic system, chemotaxis system influenced by cancer Cite This Article: Gang Li, Hui Min Hu, Xi Chen, and Fei Da Jiang, “Existence and Uniqueness of a Chemotaxis System Influenced by Cancer Cells.” International Journal of Partial Differential Equations and Applications, vol. 7, no. 1 (2020): 1-7. doi: 10.12691/ijpdea-7-1-1.

Received September 24, 2019; Revised October 29, 2019; Accepted November 15, 2019 Abstract This work introduce a difference controls of players by using a new control method to completing a pursuit game. We study pursuit game problems for controlled partial differential equations of the parabolic type. We proved a theorem on pursuit game with mixed constraints, where pursuers control are subjected to integral (geometric) constraint and geometric (integral) constraint are imposed on evaders control. Moreover, we established the sufficient conditions for which pursuit is possible in the game considered.

Keywords : hurst exponent, MATLAB, stock index, DFA. MSC, 98B28 Cite This Article: Osu B. O, Chukwunezu A. I, Olunkwa C., Obi C. N, and Okorie I. E, “Analyzing the Stock Market Using the Solution of the Fractional Option Pricing Model.” International Journal of Partial Differential Equations and Applications, vol. 6, no. 1 (2019): 1-12. doi: 10.12691/ijpdea-6-1-1.

Keywords : rangaig transform, integral transform, linear ordinary dierential function, integro-dierential equation, convolution theorem Cite This Article: Norodin A. Rangaig, Norhamida D. Minor, Grema Fe I. Pe~nonal, Jae Lord Dexter C. Filipinas, and Vernie C. Convicto, “On Another Type of Transform Called Rangaig Transform.” International Journal of Partial Differential Equations and Applications, vol. 5, no. 1 (2017): 42-48. doi: 10.12691/ijpdea-5-1-6.

Most of the important partial differential equations (PDEs) has its origin in physics and geometry. Applications of PDEs go in to many different areas of mathematical physics. Its connections with other branches of mathematics are many. One example is the work of Petrowsky in 1945 on support of fundamental solutions of hyperbolic operators with constant coefficients. Petrowsky gave a necessary and sufficient conditions for stable lacunas in terms of the homology of algebraic hyper surface given by the symbol of the operator. This work was corrected, clarified and generalized after more than twenty five years, by Atiyah, Bott and Garding [31, 32] in their two papers. The theory of integrable systems have applications in algebraic geometry. Application of heat equation in the proof of index theorems is well known. The spectral theory of Laplace-Beltrami operators and scattering theory for wave equations are used in the study of automorphic forms in number theory.

Another important distinction is between explicit and implicit methods, where the latter methods can solve a particular class of equations (so-called “stiff” equations) where explicit methods have problems with stability and efficiency. Stiffness occurs for in- stance if a problem has components with different rates of variation according to the independent vari- able. Very often there will be a tradeoff between us- ing explicit methods that require little work per inte- gration step and implicit methods which are able to take larger integration steps, but need (much) more work for one step.

This innocuous-seeming problem is suddenly pathological because it would take an infinite power source to maintain this temperature difference. Why should that be? Look at the corners. You’re trying to maintain a non-zero temperature difference ( T 0 − 0) between two walls that are touching. This can’t happen, and the equations are telling you so! It means that the boundary conditions specified in Eq. ( 10.22 ) are impossible to maintain. The temperature on the boundary at y = b can’t be constant all the way to the edge. It must drop off to zero as it approaches x = 0 and x = a . This makes the problem more difficult, but then reality is typically more complicated than our simple, idealized models. Does this make the solution Eq. ( 10.26 ) valueless? No, it simply means that you can’t push it too hard. This solution will be good until you get near the corners, where you can’t possibly maintain the constant-temperature boundary condition. In other regions it will be a good approximation to the physical problem.

. These are the equilibrium solutions, and both of them are isolated. Remarks: (i) Finding equilibria of nonlinear systems is usually impossible to do algebraically. There is no straightforward extension of the matrix algebra techniques you learned for handling linear, constant-coefficient systems. (ii) Whereas matrix systems Ax = b have either 0, 1, or infinitely many solutions, nonlinear systems can have any number of solutions. In the above example, there were two equilibria (which could never happen for the linear systems discussed in the previous chapter). As we might expect, if we start out in equilibrium of a system x 0 = f (x), then we are stuck there forever. More exactly, suppose the initial condition is x(0) = x ∗ , where x ∗ is an equilibrium. Clearly the constant function x(t) = x ∗ is a solution of the initial value problem, because x 0 = 0 and f (x) = 0. Assuming that all partial derivatives in Jf (x) are continuous in the vicinity of x ∗ , then the Fundamental Existence and Uniqueness Theorem 3.2.1 guarantees that the initial value problem has a unique solution in some interval containing t = 0. Since we have already produced the solution x(t) = x ∗ , this constant solution must be the only solution.