| In recent years, The field of physics, chemistry, biology, medicine, economics, engineering, control theory, etc. have many nonlinear problems, in order to study these nonlinear problems, people often have to resort to them the mathematical Model - the nonlinear equations. A lot of practical problems often require such nonlinear equations to satisfy one (or a group) specific conditions, such specific conditions as boundary conditions, the most important conditions is the boundary value problems.In classical mechanics and opto-electronics, the differential equations has a very rich source. From non-Newtonian flow, the physical Science, porous medium, the gas turbulence, elasticity theory, plasma problems, a large number of practical physics and on the radial solutions of nonlinear partial differential equations study people found that these problems can often be attributed to boundary value problem of ordinary differential equations. In the abstract dynamical system, before branches and chaos to become the research hotspot the qualitative theory of differential equations, Stability theory and boundary value problems for a long period of time is an important branch of differential equations. Differential equations on the one hand, because of the close ties practical problems, on the other as proof and solving methods of promotion whiich is still a concern and the new results of wide research emerging neighborhood. In practical problems, the differential equation models are often nonlinear, which makes the studying of solution of differential equations boundary value problem is particularly important. In general, the means of solving nonlinear differential equations boundary value problem with qualitative analysis, analytical solution, numerical solution and approximate solution and so on. It is undeniable that the rapid development of the computer today, the numerical method of calculation is important for solving nonlinear differential equations boundary value problem one way. This paper is a study from two aspects of the numerical solution of nonlinear boundary value problems -application of homotopy method for solving nonlinear differential equations and applications homotopy perturbation renormalization group method for solving differential equations boundary value problems. In the second chapter, we apply the homotopy method to study the boundary value problem of nonlinear differential equations, and gives solutions to solve the boundary value problem homotopy method. We are in the nonconvex domainΩ0, construct Liapunov function. We do not make convexity constraints, and then the use of homotopy continuation method to track the path starting from the initial value. finally found the solution to meet the boundary conditions. Through a number of numerical examples show that the method is effective. In the third chapter, we apply the homotopy perturbation method with renormalization group method is proposed homotopy perturbation renormalization group method, the further development of the homotopy perturbation method. This method effectively overcome the solutions of non-consistency, the method can more effectively deal with long-term items and boundary layer problems such as non-uniform. By a numerical example, we will explain it, including multi-scale, matching technical difficulties with the asymptotic boundary layer, WKB analysis of differential equations in applications. The following is our major results of article.Homotopy Method for Solving Nonlinear Boundary Value Prob- lemsConsider the first-order differential equations boundary value problem.where f : R×Rn→Rn is continous, regarding the second variable twice continuously differentiable. B : C(R, Rn)→Rn is twice continuously differentiable function, mapping bounded sets into bounded sets.We make the following assumptions.(H1) There is C2 function g : Rn×[0,1]→R, allow collection ofis a non-open set.(H2) There is C2 function (?), when y→∞, forψ(y, x0,λ)→∞,x∈(?)λ,λ∈[0,1] consistent set andψ(0, x0,λ) = 0.(H3)There is (?), when y > 0,Where (?) is(H4)For any (?), when y > 0,(H5)For any d∈Sr(0) = {q∈Rn : |q| < r}, |d| << 1,(2.1.1)1 have the only solution x(t), x(0)∈Ω1 satisfied B1(x, 1) = d.Consider the homotopy map. where (?).We have the following main results.Theory 1 Suppose (H1)-(H5) satisfy. Then for almost every (?),tracking homotopy equation (3), have a C1 path (p(s),λ(s)). such thatandλ(s)→0时, p(s) tend (x*,y*), and B(x(·, x*, 0),0)= 0.We illustrate Theorem 1. In some existing results, the use of some fixed point theorem, such as the Brouwer fixed point theorem. Liapunov functions are usually required to meet the appropriate convexity conditions, but these conditions limit the Liapunov second method of application, we removed the results of such convex constraints. In the many-oriented functions of important results, considering the guiding function is generally independent of time and require the gradient of the function mapping a given open set on the Brouwer degree of non-zero, our results do not have those conditions. Application of Theorem 1 we give a boundary value problems for non linear differential equations large global convergence method, which is a new approach.Here is our specific algorithm.Algorithm processDerived equations using numerical methods for boundary value problem (BVP) solution, we first choose an initial point of (?) to satisfy the Cauchy problem (1)λ. We follow along the path can be found (BVP) solution of p* = (x*, y*). specific algorithm is as follows:(1)Initial selection (?) andε;(2)Calculation (?); (a)When (?) ,then pois the initial value for the desired point.(b)When (?), then (3);(3) Calculation the tangent Sk of (pk,λk) ;(4)Tangent vector to take the step factor Vk;(5)Derive estimates ((?)k,λk+1);(6)Fixλk+1, at (?)k point, calculation the Newton initial directionηk of H(p0,p,λk+1)=0;(7)Take Newton step hk;(8)Seek correction point (pk+1,λk+1);(9)Calculation (?):(a)When (?), then pk+1 is the desired initial point .(b)When (?),then (3).Thus we have obtained using the initial point, we can find equation (BVP) solution. In the paper we give a concrete example.Homotopy perturbation method of renormalization groupConsider the general nonlinear equationswhere N is nonlinear operator.For the above equation, the application of homotopy analysis method (HAM) or the homotopy perturbation method (HPM) can usually obtain the solution of formBut for some of the problems of this type show the approximate solution may be non-consistent. Some sources of non-uniformity: Unlimited domains, a small parameter multiplied with the most advanced derivative, partial differential equation type of change and the existence of singularities. In the infinite domain case, the performance of non-uniform items, such as the existence of the so-called long-term tn cos t and tn sin t, It makes um(t)/um-1(t) when t tends to become unbounded infinite. In the small parameter multiplied with the most advanced derivative when the perturbation expansion does not satisfy all the boundary and initial conditions, Therefore expansion in the boundary layer and the initial layer of failure. As the partial differential equations posed problems on the boundary and initial conditions considered in the equation depends on the type of When the perturbed equation of the type with the original equation of the type is not at the same time, it may produce non-uniformity. In this case, non-exact part of the solution singularity appears in the expansion of certain points, in general, in the follow-up entry singularity becomes more pronounced.To overcome the above-mentioned non-uniformity to obtain exact solution of the problem of the uniformly valid asymptotic exhibition-style, HAM solution with the different forms of expression construct a nd then qualify, this will cause non-uniform part of the factor to To avoid non-uniform generation. But the application of HAM, solution forms of expression restricted, especially for the unknown solution of the problem, construct them was not easy. The HPM using improved Lindstedt-Poincar e perturbation techniques, However, Lindstedt-Poincar e perturbation technique of limited scope, As they seek periodic solutions to the problem is very effective, but can not seek transient response.Solve the above problem, we have proposed homotopy perturbation renormalization group method, thus the further development of the homotopy perturbation method, but also effectively overcome the knowledge of non-uniformity, the method can effectively Dealing with long-term items and boundary issues such as non-uniform. By a numerical example, we will explain it, including multi-scale, matching technical difficulties with the asymptotic boundary layer, WKB analysis of differential equations in applications.Our main results are as follows. Take account of general nonlinear systemswhereεis a small parameter, indicated time t derivative.We make the following assumptions(A1) System (3.2.1) has solution under the condition (3.2.2);(A2) f is the Analytic function u and u.Theorem 2 Suppose (A1),(A2) satisfy, u(t) is the solution of equation(3.2.1). thenwhereis Fourier coefficient, R = R(t,R(0),ε),θ=θ(t,θ(0),ε) is the renormalization group equation,R(0),θ(0) is determined by the initial conditions.Application of Theorem 2, we considered the Rayleigh equation, by solving graphical comparison can be seen in this paper obtained by the method of approximate solution and reference solution fit very well. We have also considered the following equationwhereεis a small Parameter, is derivative of t. If b/a > 0, Balance of the main items of analysis shows that t = 0 near the boundary layer exists, if b/a < 0, in t = 1 near the boundary layer exists. To simplify we only consider the b/a > 0 situation.We make the following assumptions(B1)System (3.3.2) in the boundary conditions (3.3.3)has solution, the broken;(B2) f is u of analytic functions;(B3) b/a > 0.Theorem 3 Suposse(B1)-(B3) satisfy, u(t)is the solution of (3.2.2). thenwhere (?) is the renormalization group equation,AR(0), BR(0) is define by the boundary condition (3.3.3).Application of Theorem 3, we considered the Linear and nonlinear boundary layer problem, by solving graphical comparison can be seen in this paper obtained by the method of approximate solution and reference solution fit very well.Then we consider the following system whereεis a small Parameter.We make the following assumptions(C1) The system (3.4.1) has the solution under the boundary condition(3.4.2);(C2) Q has isolation zero atx = 0, Q(x) = xαφ(x), whereφis a positive function.Theorem 4 suppose(C1),(C2)satisfy, u(x)is the solution of the equation(3.4.1). thenwhere Ai and Bi is Airy function,t(x) = (?), C = C(t, C(0),ε) is the renormalization group equationis solution, C(0) is define by condition(3.4.2).Application of Theorem 4, we considered the Time-dependent elastic constants of vibration system, by solving graphical comparison can be seen in this paper obtained by the method of approximate solution and reference solution fit very well.End of this article, we use the implicit function theorem, the homotopy perturbation renormalization group method of the renormalization group equations to simplify the formula and its application to the numerical calculation.With the slowly-varying scale AR, BR Were replaced by homotopy perturbation renormalization group method the zero-order equation contains arbitrary constants A. B, for the renormalization group transformation One sequence of renormalization constants (?),(?) forεStepwise elimination-style show in a long-term solution to determine entry. The renormalization group transformation (3.5.6) to determine the implicit functionAccording to the implicit function theorem, finally be simplified renormalization group (RG) equation forApplication of the simplified homotopy perturbation renormalization group method, we considered the Duffing equation and non-time-dependent Schr(?)dinger equation, which can show that our approach is effective.
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