Exact solutions of the generalized time-fractional order Burgers-Huxley equation with a robust semi-analytical technique

In this study, the Quasi-Linearization Method (QLM) is combined with the collocation method, based on the bivariate generalized fractional order of the Chebyshev functions (B-GFCFs), to solve higher order nonlinear partial differential equations (NPDEs) of the evolution parabolic kind with the time-arbitrary (integer or fractional) order, where are very important because the description of various mechanisms in physics and engineering. The method is used for solving some famous NPDEs, such as the generalized Burgers–Fisher equation, the Fisher equation, the Burgers–Huxley equation, and the modified KdV–Burgers equation. First, the QLM converts the nonlinear equation to a sequence of linear partial differential equations (LPDEs), and then these LPDEs are solved using the B-GFCFs collocation method. In the time-fractional case, we use the Caputo fractional derivative operational matrix to get rid of the fractional derivative. An algorithm is presented by examining its complexity order for the method, as also, the convergence analysis, error analysis, and error estimate for the method are briefly investigated. The results in the examples are compared with the exact solutions and the methods are used in other literature to show accuracy, convergence, and effectiveness of the proposed method, and also the very good approximation solutions are obtained.

2-Color Cover: This Plate PMS 296 (Blue) This Plate PMS 195 (Maroon) 320 Pages on S 40 lb Stock · 1/2 Spine · Trim 7 × 10 This book is the first of two volumes which contain the proceedings of the Workshop on Nonlinear Partial Differential Equations, held from May 28–June 1, 2012, at the University of Perugia in honor of Patrizia Pucci’s 60th birthday. The workshop brought together leading experts and researchers in nonlinear partial differential equations to promote research and to stimulate interactions among the participants. The workshop program testified to the wide ranging influence of Professor Pucci on the field of nonlinear analysis and partial differential equations. In her own work, Patrizia Pucci has been a seminal influence in many important areas: the maximum principle, qualitative analysis of solutions to many classes of nonlinear PDEs (Kirchhoff problems, polyharmonic systems), mountain pass theorem in the critical case, critical exponents, variational identities, as well as various degenerate or singular phenomena in mathematical physics. This same breadth is reflected in the mathematical papers included in this volume. The companion volume (Contemporary Mathematics, Volume 595) is devoted to stationary problems in nonlinear partial differential equations.

Operators with regular singularities. One variable case - Operators with regular singularities. Several variables case - Formal and convergent solutions of singular partial differential equations - Local study of differential equations of the form xy' f(x,y) near x 0 - Holomorphic and singular solutions of non linear singular first order partial differential equations - Maillet's type theorems for non linear singular partial differential equations - Maillet's type theorems for non linear singular partial differential equations without linear part - Holomorphic and singular solutions of non linear singular partial differential equations - On the existence of holomorphic solutions of the Cauchy problem for non linear partial differential equations - Maillet's type theorems for non linear singular integro-differential equations.

In this research work, we propose an algorithm which involves the coupling of a new integral transform called the Elzaki transform and the well-known homotopy perturbation method on the Burgers–Huxley equation which is a type of nonlinear advection–diffusion partial differential equation. The Burgers–Huxley equation which models reaction mechanisms, diffusion transports and nerve ion propagation which is applicable in traffic flows, acoustics, turbulence theory, hydrodynamics and generally mechanics is the fusion of the well-known Burgers equation and the Huxley equation. We proffer an analytical solution in the form of a Taylor multivariate series of displacement x and time t using the proposed Elzaki homotopy transformation perturbation method (EHTPM) to three cases of the Burgers–Huxley equation. This solution converges rapidly to a closed form which is the same as the exact solutions obtained using the normal analytical methods from the existing literatures. The exact results and that of our proposed EHTPM when compared via tables and 3D plots show an excellent agreement devoid of errors.

In this paper, modified simple equation method has been applied to ob-tain generalized solutions of Burgers, Huxley equations and combined forms of these equations. The new exact solutions of these equations have been obtained. It has been shown that the proposed method provides a very effective, and powerful mathematical tool for solving nonlinear partial differential equations.

In this note, we discuss symmetry properties of solutions for simple scalar and vector-valued systems of nonlinear elliptic partial differential equations (PDEs). The systems of interest are of variational nature, they appear, for instance, in some first-order phase transition models in mathematical physics (e.g., phase separation, superconductivity, liquid crystals) and are naturally related to some PDEs in geometry (minimal surfaces and harmonic maps). We review some known results and present few open problems about symmetry of minimizers for all these models.

Solving and learning nonlinear PDEs with Gaussian processes

This work focused on developing and analyzing semi analytical scheme that is a computer-based approach for solving complex evolutionary partial differential equations (PDEs) of the Burger type. A novel scheme named Asymptotic Homotopy Perturbation Transform Method (AHPTM) is introduced for solving fractional order coupled nonlinear Burgers PDEs. The Caputo fractional form has been considered for derivatives. Nonlinear Burgers PDEs have various applications in fluid dynamics, traffic flow, nonlinear acoustics and signal processing. The algorithm of AHPTM is a fast convergent approach that has been developed by combining the Laplace transformation and the asymptotic homotopy perturbation method. Using the technique of AHPTM, three test problems of one-dimensional coupled nonlinear Burgers PDEs have been solved. This work demonstrates the smooth and easy implementation of solving problems by using MATLAB programming. The numerical results demonstrate that this novel approach is simple, easy and computationally capable for the problems. An error estimate is also required to solve the problem. To demonstrate the effectiveness and accuracy of the said solver, the solutions to the problems are tabulated and graphically displayed via using MATLAB software.

The Goursat partial differential equation arises in linear and non linear partial differential equations with mixed derivatives. This equation is a second order hyperbolic partial differential equation which occurs in various fields of study such as in engineering, physics, and applied mathematics. There are many approaches that have been suggested to approximate the solution of the Goursat partial differential equation. However, all of the suggested methods traditionally focused on numerical differentiation approaches including forward and central differences in deriving the scheme. An innovation has been done in deriving the Goursat partial differential equation scheme which involves numerical integration techniques. In this paper we have developed a new scheme to solve the Goursat partial differential equation based on the Adomian decomposition (ADM) and associated with Boole-s integration rule to approximate the integration terms. The new scheme can easily be applied to many linear and non linear Goursat partial differential equations and is capable to reduce the size of computational work. The accuracy of the results reveals the advantage of this new scheme over existing numerical method.

In this paper, variational iteration method is applied to compute the numerical solutions of non-linear partial differential equations like Huxley and Burgers–Huxley equation. The approximate solutions of the Huxley and Burgers–Huxley equations are compared with the Haar wavelet solution as well as with the exact solutions. The present method is very simple, effective and convenient analytical method with small computational overhead.

On Stochastic Processes Defined by Differential Equations with a Small Parameter

The numerical methods are of great importance for approximating the solutions of nonlinear ordinary or partial differential equations, especially when the nonlinear differential equation under consideration faces difficulties in obtaining its exact solution. In this latter case, we usually resort to one of the efficient numerical methods. In this paper, the Chebyshev collocation method is suggested to deal numerically with some nonlinear partial differential equations in mathematical physics.

In this paper, we have applied the mapping method to solve the (3+1)-dimensional Boussinenesq equation where we have obtained exact solutions for evolution equation to construct exact periodic and soliton solutions of nonlinear partial differential evolution equation. Many have obtained new families of exact traveling wave solutions, but the Boussinesq equation is successfully. These solutions may be significantly important for the explanation of some practical physical problems. New exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions and elliptic functions. It is shown that the mapping method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.

In the presence of transaction costs the perfect option replication is impossible which invalidates the celebrated Black and Scholes (1973) model. In this chapter we consider some approaches to option pricing and hedging in the presence of transaction costs. The distinguishing feature of all these approaches is that the solution for the option price and hedging strategy is given by a nonlinear partial differential equation (PDE). We start with a review of the Leland (1985) approach which yields a nonlinear parabolic PDE for the option price, one of the first such in finance. Since the Leland's approach to option pricing has been criticized on different grounds, we present a justification of this approach and show how the performance of the Leland's hedging strategy can be improved. We extend the Leland's approach to cover the pricing and hedging of options on commodity futures contracts, as well as path-dependent and basket options. We also present examples of finite-difference schemes to solve some nonlinear PDEs. Then we proceed to the review of the most successful approach to option hedging with transaction costs, the utility-based approach pioneered by Hodges and Neuberger (1989). Judging against the best possible tradeoff between the risk and the costs of a hedging strategy, this approach seems to achieve excellent empirical performance. The asymptotic analysis of the option pricing and hedging in this approach reveals that the solution is also given by a nonlinear PDE. However, this approach has one major drawback that prevents the broad application of this approach in practice, namely, the lack of a closed-form solution. The numerical computations are cumbersome to implement and the calculations of the optimal hedging strategy are time consuming. Using the results of asymptotic analysis we suggest a simplified parameterized functional form of the optimal hedging strategy for either a single option or a portfolio of options and a method for finding the optimal parameters.

We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1 + 1)-dimensional hierarchy of nonlinear PDEs linearizable by the matrix Hopf–Cole substitution (the Bürgers hierarchy). We derive examples of four-dimensional nonlinear matrix PDEs together with the scalar and three-dimensional reductions. Variants of the Kadomtsev–Petviashvili-type and Korteweg–de Vries-type equations are represented among them. Our algorithm is based on the combination of two Frobenius-type reductions and special differential reduction imposed on the matrix fields of integrable PDEs. It is shown that the derived four-dimensional nonlinear PDEs admit arbitrary functions of two variables in their solution spaces which clarifies the integrability degree of these PDEs.