The Order Completion Method for Systems of Nonlinear PDEs Revisited

Kranc: a Mathematica package to generate numerical codes for tensorial evolution equations

A frozen Jacobian iterative method is proposed for solving systems of nonlinear equations. In particular, we are interested in solving the systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). In a single instance of the proposed iterative method DEDF, we evaluate two Jacobians, one inversion of the Jacobian and four function evaluations. The direct inversion of the Jacobian is computationally expensive, so, for a moderate size, LU factorization is a good direct method to solve the linear system. We employed the LU factorization of the Jacobian to avoid the direct inversion. The convergence order of the proposed iterative method is at least eight, and it is nine for some particular classes of problems. The discretization of IVPs and BVPs is employed by using Jacobi-Gauss-Lobatto collocation (J-GL-C) method. A comparison of J-GL-C methods is presented in order to choose best collocation method. The validity, accuracy and the efficiency of our DEDF are shown by solving eleven IVPs and BVPs problems.

A rigorous methodology for the analysis of initial boundary value problems on the half-line, 0 < x < ∞, t > 0, for integrable nonlinear evolution PDEs has recently appeared in the literature. As an application of this methodology the solution q(x, t) of the sine-Gordon equation can be obtained in terms of the solution of a 2 × 2 matrix Riemann–Hilbert problem. This problem is formulated in the complex k-plane and is uniquely defined in terms of the so-called spectral functions a(k), b(k), and B(k)/A(k). The functions a(k) and b(k) can be constructed in terms of the given initial conditions q(x, 0) and qt(x, 0) via the solution of a system of two linear ODEs, while for arbitrary boundary conditions the functions A(k) and B(k) can be constructed in terms of the given boundary condition via the solution of a system of four nonlinear ODEs. In this paper, we analyse two particular boundary conditions: the case of constant Dirichlet data, q(0, t) = χ, as well as the case when qx(0, t), sin (q(0, t)/2), and cos(q(0, t)/2) are linearly related by two constants χ1 and χ2. We show that for these particular cases, the system of the above nonlinear ODEs can be avoided, and B(k)/A(k) can be computed explicitly in terms of { a(k), b(k), χ} and {a(k), b(k), χ1, χ2}, respectively. Thus, these ‘linearizable’ initial boundary value problems can be solved with absolutely the same level of efficiency as the classical initial value problem of the line.

Contrary to widespread perception, there is ever since 1994 a unified, general type independent theory for the existence of solutions for very large classes of nonlinear systems of PDEs.This solution method is based on the Dedekind order completion of suitable spaces of piece-wise smooth functions on the Euclidean domains of definition of the respective PDEs.The method can also deal with associated initial and/or boundary value problems.The solutions obtained can be assimilated with usual measurable functions or even with Hausdorff continuous functions on the respective Euclidean domains.It is important to note that the use of the order completion method does not require any monotonicity condition on the nonlinear systems of PDEs involved.One of the major advantages of the order completion method is that it eliminates the algebra based dichotomy "linear versus nonlinear" PDEs, treating both cases with equal ease.Furthermore, the order completion method does not introduce the dichotomy "monotonous versus non-monotonous" PDEs.None of the known functional analytic methods can exhibit such a performance, since in addition to topology, such methods are significantly based on algebra.

The shifted Chebyshev polynomial approximation is employed to solve the linear, constant parameter, ordinary differential equations of initial or two-point boundary value problems. An effective recursive algorithm is developed to calculate the expansion coefficients of the shifted Chebyshev series. An effective transformation is proposed to transform the two-point boundary value problem into an initial value problem. An illustrative example is included to show that the computational results are accurate.

We prove local existence, uniqueness, Holder regularity in space and time, and smooth dependence in Holder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, which is necessary for the applicability of differential calculus to abstract formulations of the initial boundary value problems, has been closed. The theory works for any space dimension, and the nonlinearities in the equations as well as in the boundary conditions are allowed to be nonlocal and to have any growth. The main tools are new maximal regularity results (Griepentrog in Adv Differ Equ 12:781–840, 1031–1078, 2007) in Sobolev–Morrey spaces for linear parabolic initial boundary value problems with nonsmooth data, linearization techniques and the Implicit Function Theorem.

This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, u=u(x,t), these equations contain the same function at a past time, w=u(x,t−τ), where τ&gt;0 is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments, w=u(px,qt), where p and q are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.

Numerical solution of two-point boundary value problems

Global existence of smooth solutions is proved for an effective theory of bubbly liquids for either the initial value problem or initial boundary value problem in one dimension. This shows that the theory does not describe shock waves or bubble collapse. Since the analysis is not for the steady boundary value problem, there is no discussion of resonance. The proof uses a semilinear form of the equations to get local existence. A priori bounds resulting from energy conservation and a nonlinear Gronwall‐like inequality are then derived to prove global existence.

&lt;p&gt;In this paper, we present a framework to determine Haar solutions of Bratu-type equations that are widely applicable in fuel ignition of the combustion theory and heat transfer. The method is proposed by applying Haar series for the highest derivatives and integrate the series. Several examples are given to confirm the efficiency and the accuracy of the proposed algorithm. The results show that the proposed way is quite reasonable when compared to exact solution.&lt;/p&gt;

A computer program is described which has been developed for obtaining approximate solutions to linear initial and boundary value problems involving differential equations. The program can be used for ordinary curve fitting as well. For each problem, input to the program includes:1. The equations (in symbolic form) to be satisfied - the differential equations, equations describing auxiliary conditions such as boundary conditions, etc.2. A numerical description of the regions in which each of the equations are to be satisfied.3. Sets of functions (in symbolic form) to be used in linear combinations to approximate the solution functions.Given the above input, the program generates an approximation to the solutuon of the specified problem in terms of the specified functions which is optimum in the least squares sense. A number of examples illustrating the use of the program are included in the paper.

On initial value and terminal value problems for Hamilton–Jacobi equation

Boundary value problems (BVP) arise in many mathematical model of frontier science. In short, a BVP is a second order ordinary differential equation with boundary conditions prescribed at two distinct points. Usually, we prescribe three types of boundary conditions: Dirichlet, Neumann and Mixed. Among them mixed boundary conditions is little difficult to approach. One of the method to solve the BVP with mixed boundary conditions is the shooting method. In this method, we first convert the BVP into an initial value problems (IVP) and then solve it. We then check whether the solution of IVP is a solution of the BVP. There are standard ways of solving IVPs. Recently, some alternate methods are also used to solve them. For example, several authors use Haar wavelet. Frames are considered as an extension to wavelet. In this paper, we construct operational matrices from a frame and then apply them to solve the IVP which is obtained from the given BVP by shooting method. This method is simple and easy to use because we construct the operational matrix once, store it and then apply it any times we want. Moreover, we also show that the given operational matrix method converges quadratically.

An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann–Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow‐up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow‐up of a solution in a finite time in a limited area with a nonlinear Neumann–Dirichlet boundary condition are obtained.

In the literature so far, for Caputo fractional boundary value problems of order 2q when 1&lt;2q&lt;2, the problems use the same boundary conditions of the integer-order differential equation of order ‘2’. In addition, they only use the left Caputo derivative in computing the solution of the Caputo boundary value problem of order 2q. Further, even the initial conditions for a Caputo fractional differential equation of order nq use the corresponding integer-order initial conditions of order ‘n’. In this work, we establish that it is more appropriate to use the Caputo fractional initial conditions and Caputo fractional boundary conditions for sequential initial value problems and sequential boundary value problems, respectively. It is to be noted that the solution of a Caputo fractional initial value problem or Caputo fractional boundary value problem of order ‘nq’ will only be a Cnq solution and not a Cn solution on its interval. In this work, we present a methodology to compute the solutions of linear sequential Caputo fractional differential equations using initial and boundary conditions of fractional order kq, k=0,1,…(n−1) when the order of the fractional derivative involved in the differential equation is nq. The Caputo left derivative can be computed only when the function can be expressed as f(x−a). Then the Caputo right derivative of the same function will be computed for the function f(b−x). Further, we establish that the relation between the Caputo left derivative and the Caputo right derivative is very essential for the study of Caputo fractional boundary value problems. We present a few numerical examples to justify that the Caputo left derivative and the Caputo right derivative are equal at any point on the Caputo function’s interval. The solution of the linear sequential Caputo fractional initial value problems and linear sequential Caputo fractional boundary value problems with fractional initial conditions and fractional boundary conditions reduces to the corresponding integer initial and boundary value problems, respectively, when q=1. Thus, we can use the value of q as a parameter to enhance the mathematical model with realistic data.