Dynamical Properties of Periodic Solutions of Integro-Differential Equations

We consider the general system of n first order linear ordinary differential equations y'(t)=A(t)y(t)+g(t), a<t< b, subject to "boundary" conditions, or rather linear constraints, of the form Σ^(N)_(ν=1) B_(ν)y(τ_ν)=β Here y(t), g(t) and II are n-vectors and A(t), Bx,..., BN are n × n matrices. The N distinct points {τ_ν} lie in [a,b] and we only require N ≧ 1. Thus as special cases initial value problems, N=1, are included as well as the general 2-point boundary value problem, N=2, with τ_1=a, τ_2=b. (More general linear constraints are also studied, see (5.1) and (5.17).)

This paper is concerned with the nonlinear boundary value problem (1) $\beta u''-u'+f(u)=0$, (2) $u'(0)-au(0)=0,u'(1)=0$, where $f(u)=b(c-u)\exp(-k/(1+u))$ and $\beta,a,b,c,k$ are constants. First a formal singular perturbation procedure is applied to reveal the possibility of multiple solutions of (1) and (2). Then an iteration procedure is introduced which yields sequences converging to the maximal solution from above and the minimal solution from below. A criterion for a unique solution of (1), (2) is given. It is mentioned that for certain values of the parameters multiple solutions have been found numerically. Finally, the stability of solutions of (1), (2) is discussed for certain values of the parameters. A solution $u(x)$ of (1), (2) is said to be stable if the first eigenvalue $\sigma$ of the variational equations $(1)' \beta v''-v'+[\sigma\beta+f'(u)]v=0$ and $(2)' v'(0)-av(0)=0, v'(1)=0$, is positive.

Previous article Next article Numerical Solution of Two-Point Boundary Value Problems on Total Differential EquationsDavid GlasserDavid Glasserhttps://doi.org/10.1137/0706054PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Henry J. Kelley, G. Leitman, Method of gradientsOptimization techniques, Academic Press, New York, 1962, 205–254, Chap. 6. MR0162671 CrossrefGoogle Scholar[2] L. Fox, Numerical solution of ordinary and partial differential equations., Based on a Summer School held in Oxford, August-September 1961, Pergamon Press, Oxford, 1962ix+509 MR0146969 0101.09904 Google Scholar[3] Lothar Collatz, Functional analysis and numerical mathematics, Translated from the German by Hansjörg Oser, Academic Press, New York, 1966xx+473 MR0205126 0148.39002 Google Scholar[4] E. S. Lee, Quasi-linearization, non-linear boundary value problems and optimization, Chem. Eng. Sci., 21 (1966), 183–194 10.1016/0009-2509(66)85007-8 CrossrefISIGoogle Scholar[5] F. H. Deist and , L. Sefor, Solutions of systems of non-linear equations by parameter variation, Comput. J., 10 (1967), 78–82 10.1093/comjnl/10.1.78 0171.35603 CrossrefISIGoogle Scholar[6] C. B. Haselgrove, The solution of non-linear equations and of differential equations with two-point boundary conditions, Comput. J., 4 (1961/1962), 255–259 10.1093/comjnl/4.3.255 MR0130114 0121.11301 CrossrefISIGoogle Scholar[7] O. Vejvoda, I. Babushka, Nonlinear boundary-value problems for differential equations, Differential Equations and Their Applications (Proc. Conf., Prague, 1962), Publ. House Czechoslovak Acad. Sci., Prague, 1963, 199–215 MR0173072 0196.40102 Google Scholar[8] Thomas L. Saaty and , Joseph Bram, Nonlinear mathematics, McGraw-Hill Book Co., New York, 1964xv+381 MR0164812 0198.00102 Google Scholar[9] L. Lapidus, Digital Computations for Chemical Engineers, McGraw-Hill, New York, 1962 Google Scholar[10] E. S. Lee, Optimization by Pontryagin's maximum principle on the analog computer, American Inst. Chem. Eng. J., 10 (1964), 309–315 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A Continuation Method for Nonlinear RegressionNoël de Villiers and David GlasserSIAM Journal on Numerical Analysis, Vol. 18, No. 6 | 17 July 2006AbstractPDF (1889 KB)Numerical solution of nonlinear equations by one-parameter imbedding methodsNumerical Functional Analysis and Optimization, Vol. 3, No. 2 | 26 June 2007 Cross Ref Suboptimal control of systems with multiple delaysJournal of Optimization Theory and Applications, Vol. 30, No. 4 | 1 Apr 1980 Cross Ref Solution of nonlinear boundary value problems—XIChemical Engineering Science, Vol. 34, No. 5 | 1 Jan 1979 Cross Ref One-parameter imbedding techniques for the solution of nonlinear boundary-value problemsApplied Mathematics and Computation, Vol. 4, No. 4 | 1 Oct 1978 Cross Ref Optimal control of nonlinear power systems by an imbedding methodAutomatica, Vol. 11, No. 6 | 1 Nov 1975 Cross Ref NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS: SURVEY AND SOME RECENT RESULTS ON DIFFERENCE METHODSNumerical Solutions of Boundary Value Problems for Ordinary Differential Equations | 1 Jan 1975 Cross Ref On the optimal control solution of nonlinear power systemsIFAC Proceedings Volumes, Vol. 7, No. 2 | 1 Oct 1974 Cross Ref Generalized Quasi-Einearization MethodAIAA Journal, Vol. 12, No. 9 | 1 Sep 1974 Cross Ref Parameter variation for the solution of two-point boundary-value problems and applications in the calculus of variationsJournal of Optimization Theory and Applications, Vol. 13, No. 2 | 1 Feb 1974 Cross Ref Extension of a perturbation technique for nonlinear two-point boundary-value problemsJournal of Optimization Theory and Applications, Vol. 12, No. 5 | 3 August 2013 Cross Ref An indirect trajectory optimization algorithm based on the continuation method for solution of nonlinear equationsGuidance and Control Conference | 16 August 1973 Cross Ref The epsilon variation method in two-point boundary-value problemsJournal of Optimization Theory and Applications, Vol. 12, No. 2 | 3 August 2013 Cross Ref Numerical Solutions by the Continuation MethodE. WasserstromSIAM Review, Vol. 15, No. 1 | 18 July 2006AbstractPDF (2435 KB)A manifold imbedding algorithm for optimization problemsAutomatica, Vol. 8, No. 5 | 1 Sep 1972 Cross Ref On the imbedding solution of a class of optimal control problemsAutomatica, Vol. 8, No. 5 | 1 Sep 1972 Cross Ref A Manifold Imbedding Algorithm for Optimization ProblemsIFAC Proceedings Volumes, Vol. 5, No. 1 | 1 Jun 1972 Cross Ref Solving Boundary-Value Problems by ImbeddingJournal of the ACM, Vol. 18, No. 4 | 1 Oct 1971 Cross Ref Volume 6, Issue 4| 1969SIAM Journal on Numerical Analysis523-616 History Submitted:02 July 1968Published online:14 July 2006 InformationCopyright © 1969 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0706054Article page range:pp. 591-597ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

Here f : R ×R → R and R : R × R → R. Throughout the following, we assume that the conditions imposed on the nonlinear functions f and R of the variable y on [a, b] provide that problem (1.1), (1.2) is solvable. Numerous methods were developed for the solution of boundary value problems of the form (1.1), (1.2). The paper [1] presents approximate methods for operator equations and the investigation of various iterative processes for linear and nonlinear equations. For the choice of an initial approximation for a nonlinear operator equation, one introduces a parameter λ ∈ [0, 1] such that if λ = 0, then the solution of the equation is known or can be readily found, and if λ = 1, then one has the solution of the original equation. Then the desired solution is constructed with the use of continuation with respect to the parameter λ. Three methods for nonlinear boundary value problems were considered in [2] : the shooting method, the method of Green function, and the finite-difference method. The monograph [3] deals with the exposition of numerical-analytic methods for nonlinear systems of differential equations considered under nonseparatable two-point boundary conditions. It was suggested to introduce a parameter in boundary conditions so as to ensure that the solution is known at the first step of the method and the solution of the original boundary value problem is obtained at the last step with respect to the parameter. The nonlinear boundary value problem was preliminarily linearized in the monograph [4]. To improve the composition method, it was suggested to perform Godunov reorthogonalization [5, 6]. To this end, it was suggested to use the Cont algorithm [7]. A universal approach (quasilinearization) to the solution of various nonlinear problems was described in [8]. Bakhvalov [9] suggested to solve a nonlinear boundary value problem (after the linearization) with the use of the Thomas differential orthogonal method [5, 6]. The monograph [10] dealt with various numerical methods for twoand many-point boundary value problems.

and Applied Analysis 3 1-homogeneous operator in a Banach space and then demonstrate its application in establishing the existence of positive solutions for p-Laplacian boundary value problems under certain conditions. (xi) In the paper titled “Existence of solutions for nonhomogeneous A-harmonic equations with variable growth,” the authors establish a theorem for the existence of weak solutions for nonhomogeneous A-harmonic equations in subspace and then give three examples to demonstrate its application. (xii) In the paper titled “Multiple solutions for degenerate elliptic systems near resonance at higher eigenvalues,” the authors study the degenerate semilinear elliptic system in an open bounded domain with smooth boundary, and some multiplicity results of solutions are obtained for the system near resonance at certain eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory. (xiii) In the paper titled “A regularity criterion for the Navier-Stokes equations in the multiplier spaces,” the authors establish a regularity criterion in terms of the pressure gradient for weak solutions to the NavierStokes equations in a special class. The third set of papers, including four papers, deal with several boundary value problems for highly nonlinear ordinary differential equations. (i) In the paper titled “Positive solutions for second-order singular semipositone differential equations involving Stieltjes integral conditions,” the authors investigate the existence of positive solutions for second-order singular differential equations with a negatively perturbed term, by means of the fixed-point theory in cones. (ii) In the paper titled “Positive solutions for Sturm-Liouville boundary value problems in a Banach Space,” the sufficient conditions for the existence of single and multiple positive solutions for a second-order SturmLiouville boundary value problem are established in a Banach space, by using the fixed-point theorem of strict set contraction operators in the frame of the ODE technique. (iii) In the paper titled “Positive solutions of a nonlinear fourth-order dynamic eigenvalue problem on time scales,” the authors study a nonlinear fourth-order dynamic eigenvalue problem on time scales and obtain the existence and nonexistence of positive solutions when 0 λ, respectively, for some λ, by using the Schauder fixed-point theorem and the upper and lower solution method. (iv) In the paper titled “Bifurcation analysis for a predatorprey model with time delay and delay-dependent parameters,” a class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. By using the normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifur-cating from Hopf bifurcations are obtained. The fourth set of papers focus on finding the approximate and numerical solutions of various complex nonlinear boundary value problems. (i) In the paper titled “On spectral homotopy analysis method for solving linear Volterra and Fredholm integrodifferential equations,” a spectral homotopy analysis method (SHAM) is proposed to solve linear Volterra integrodifferential equations, and some examples are given to test the efficiency and the accuracy of the proposed method. (ii) In the paper titled “The solution of a class of singularly perturbed two-point boundary value problems by the iterative reproducing kernel method,” the authors establish an iterative reproducing kernel method (IRKM) for solving singular perturbation problems with boundary layers and give two numerical examples to demonstrate the effectiveness of the method. (iii) In the paper titled “A Galerkin solution for Burgers’ equation using cubic B-spline finite elements,” a Galerkin method using cubic B-splines is set up to find the numerical solutions of Burgers’ equation, and the method is shown to be capable of solving Burgers’ equation accurately for values of viscosity ranging from very small to very large. (iv) In the paper titled “Forward-backward splitting methods for accretive operators in Banach spaces,” the authors introduce two iterative forward-backward splitting methods with relaxations to find zeros of the sum of two accretive operators in Banach spaces and prove the weak and strong convergence of these methods under mild conditions, and also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem. Yong Hong Wu Lishan Liu Benchawan Wiwatanapataphee Shaoyong Lai Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Differential Equations International Journal of Volume 2014 Applied Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Probability and Statistics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Physics Advances in Complex Analysis Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Operations Research Advances in

Lagrangian coordinates in free boundary problems for parabolic equations

This paper is concerned with the existence and uniqueness of solutions for boundary value problems with p-Laplacian delay differential equations on the half-line. The existence of solutions is derived from the Schauder fixed point theorem, whereas the uniqueness of solution is established by the Banach contraction principle. As an application, an example is given to demonstrate the main results.MSC:34K10, 34B18, 34B40.

CONTENTS Introduction § 1. The maximum principle. Uniqueness of the solutions of the basic boundary value problems § 2. A priori estimates § 3. Solution of boundary value problems by Rothe's method. The Cauchy problem § 4. The fundamental solution of a linear parabolic equation. The Green's function. The method of integral equations for the solution of boundary value problems § 5. Generalized solutions of boundary value problems. The uniqueness theorem. Some auxiliary propositions § 6. The method of finite differences § 7. Some methods of functional analysis for the solution of boundary value problems § 8. The solution of boundary value problems by the method of continuation by a parameter § 9. The application of Galerkin's method for the construction of a solution of the first boundary value problem § 10. Generalized solutions of Cauchy's problem § 11. On differentiability properties of generalized solutions § 12. The behaviour of solutions for indefinitely increasing time References

I. Third Order Linear Homogeneous Differential Equations in Normal Form.- 1. Fundamental Properties of Solutions of the Third Order Linear Homogeneous Differential Equation.- 1. The Normal Form of a Third Order Linear Homogeneous Differential Equation.- 2. Adjoint and Self-adjoint Third Order Linear Differential Equations.- 3. Fundamental Properties of Solutions.- 4. Relationship between Solutions of the Differential Equations (a) and (b).- 5. Integral Identities.- 6. Notion of a Band of Solutions of the First, Second and Third Kinds.- 7. Further Properties of Solutions of the Differential Equation (a) Implied by Properties of Bands.- 8. Weakening of Property (v) for the Laguerre Invariant.- 2. Oscillatory Properties of Solutions of the Differential Equation (a).- 1. Basic Definitions.- 2. Sufficient Conditions for the Differential Equation (a) to Be Disconjugate.- 3. Sufficient Conditions for Oscillatoricity of Solutions of the Differencial Equation (a).- 4. Further Conditions Concerning Oscillatoricity or Non-oscillatoricity of Solutions of the Differential Equation (a).- 5. Relation between Solutions without Zeros and Oscillatoricity of the Differential Equation (a).- 6. Sufficient Conditions for Oscillatoricity of Solutions of the Differential Equation (a) in the Case A(x) ? 0, x ? (a, ?).- 7. Conjugate Points, Principal Solutions and the Relationship between the Adjoint Differential Equations (a) and (b).- 8. Criteria for Oscillatoricity of the Differential Equations (a) and (b) Implied by Properties of Conjugate Points.- 9. Further Criteria for Oscillatoricity of the Differential Equation (b).- 10. The Number of Oscillatory Solutions in a Fundamental System of Solutions of the Differential Equation (a).- 11. Criteria for Oscillatoricity of Solutions of the Differential Equation (a) in the Case that the Laguerre Invariant Does Not Satisfy Condition (v).- 12. The Case, When the Laguerre Invariant Is an Oscillatory Function of x.- 13. The Differential Equation (a) Having All Solutions Oscillatory in a Given Interval.- 3. Asymptotic Properties of Solutions of the Differential Equations (a) and (b).- 1. Asymptotic Properties of Solutions without Zeros of the Differential Equations (a) and (b).- 2. Asymptotic Properties of Oscillatory Solutions of the Differential Equation (b).- 3. Asymptotic Properties of All Solutions of the Differential Equation (a).- 4. Boundary Value Problems.- 1. The Green Function and Its Applications.- 2. Further Applications of Integral Equations to the Solution of Boundary-value Problems.- 3. Generalized Sturm Theory for Third Order Boundary-value Problems.- 4. Special Boundary-value Problems.- II. Third Order Linear Homogeneous Differential Equations with Continuous Coefficients.- 5. Principal Properties of Solutions of Linear Homogeneous Third Order Differential Equations with Continuous Coefficients.- 1. Principal Properties of Solutions of the Differential Equation (A).- 2. Bands of Solutions of the Differential Equation (A).- 3. Application of Bands to Solving a Three-point Boundary-value Problem.- 6. Conditions for Disconjugateness, Non-oscillatoricity and Oscillatoricity of Solutions of the Differential Equation (A).- 1. Conditions for Disconjugateness of Solutions of the Differential Equation (A).- 2. Solutions without Zeros and Their Relation to Oscillatoricity of Solutions of the Differential Equation (A).- 3. Conditions for the Existence of Oscillatory Solutions of the Differential Equation (A).- 4. On Uniqueness of Solutions without Zeros of the Differential Equation (A).- 5. Some Properties of Solutions of the Differential Equation (A) with r(x) ? 0.- 7. Comparison Theorems for Differential Equations of Type (A) and Their Applications.- 1. Comparison Theorems.- 2. A Simple Application of Comparison Theorems.- 3. Remark on Asymptotic Properties of Solutions of the Differential Equation (A).- III. Concluding Remarks.- 1. Special Forms of Third Order Differential Equations.- 2. Remark on Mutual Transformation of Solutions of Third Order Differential Equations.- IV. Applications of Third Order Linear Differential Equation Theory.- 8. Some Applications of Linear Third Order Differential Equation Theory to Non-linear Third Order Problems.- 1. Application of Quasi-linearization to Certain Problems Involving Ordinary Third Order Differential Equations.- 2. Three-point Boundary-value Problems for Third Order Non-linear Ordinary Differential Equations.- 3. On Properties of Solutions of a Certain Non-linear Third Order Differential Equation.- 9. Physical and Engineering Applications of Third Order Differential Equations.- 1. On Deflection of a Curved Beam.- 2. Three-layer Beam.- 3. Survey of Some Other Applications of Third Order Differential Equations.- References.

Previous article Next article On the Transformation of a Class of Boundary Value Problems into Initial Value Problems for Ordinary Differential EquationsMurray S. KlamkinMurray S. Klamkinhttps://doi.org/10.1137/1004006PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. Goldstein, Modern Developments in Fluid Dynamics, Vol. I, Oxford, London, 1957, 135–136 Google Scholar[2] H. P. Greenspan and , G. F. Carrier, The magnetohydrodynamic flow past a flat plate, J. Fluid Mech., 6 (1959), 77–96 MR0108191 0088.19203 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A note on the transformation of boundary value problems to initial value problems: The iterative transformation methodApplied Mathematics and Computation, Vol. 415 | 1 Feb 2022 Cross Ref Sawtooth patterns in flexural force curves of structural biological materials are not signatures of toughness enhancement: Part IIJournal of the Mechanical Behavior of Biomedical Materials, Vol. 124 | 1 Dec 2021 Cross Ref Group invariant solution for a pre-existing fracture driven by a power-law fluid in permeable rockInternational Journal of Modern Physics B, Vol. 30, No. 28n29 | 20 Nov 2016 Cross Ref Free Boundary Formulation for BVPs on a Semi-infinite Interval and Non-iterative Transformation MethodsActa Applicandae Mathematicae, Vol. 140, No. 1 | 23 October 2014 Cross Ref Propagation of a pre-existing turbulent fluid fractureInternational Journal of Non-Linear Mechanics, Vol. 54 | 1 Sep 2013 Cross Ref An investigation of an Emden-Fowler equation from thin film flowActa Mechanica Sinica, Vol. 28, No. 2 | 28 February 2012 Cross Ref A group invariant solution for a pre-existing fluid-driven fracture in permeable rockNonlinear Analysis: Real World Applications, Vol. 12, No. 1 | 1 Feb 2011 Cross Ref On the equivalence of non-iterative transformation methods based on scaling and spiral groupsMathematical Methods in the Applied Sciences, Vol. 33, No. 5 | 19 June 2009 Cross Ref Analysis of the constant B-number assumption while modeling flame spreadCombustion and Flame, Vol. 152, No. 3 | 1 Feb 2008 Cross Ref Upward flame spread on a vertically oriented fuel surface: The effect of finite widthProceedings of the Combustion Institute, Vol. 31, No. 2 | 1 Jan 2007 Cross Ref Автомодельные решения и степенная геометрияУспехи математических наук, Vol. 55, No. 1 | 1 Jan 2000 Cross Ref BibliographyPower Geometry in Algebraic and Differential Equations | 1 Jan 2000 Cross Ref Self-similar solutionsPower Geometry in Algebraic and Differential Equations | 1 Jan 2000 Cross Ref A Similarity Approach to the Numerical Solution of Free Boundary ProblemsSIAM Review, Vol. 40, No. 3 | 2 August 2006AbstractPDF (387 KB)A Novel Approach to the Numerical Solution of Boundary Value Problems on Infinite IntervalsSIAM Journal on Numerical Analysis, Vol. 33, No. 4 | 12 July 2006AbstractPDF (1468 KB)Nonlinear boundary value problems on semi-infinite intervalsComputers & Mathematics with Applications, Vol. 28, No. 10-12 | 1 Nov 1994 Cross Ref Numerical transformation methods: a constructive approachJournal of Computational and Applied Mathematics, Vol. 50, No. 1-3 | 1 May 1994 Cross Ref The falkneer-skan equation: Numerical solutions within group invariance theoryCalcolo, Vol. 31, No. 1-2 | 1 Mar 1994 Cross Ref Non-iterative Transformation Methods EquivalenceModern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics | 1 Jan 1993 Cross Ref Optimal numerical algorithmsApplied Numerical Mathematics, Vol. 10, No. 3-4 | 1 Sep 1992 Cross Ref Special TopicsNumerical Methods for Partial Differential Equations | 1 Jan 1992 Cross Ref A noniterative transformation method applied to two-point boundary-value problemsApplied Mathematics and Computation, Vol. 39, No. 1 | 1 Sep 1990 Cross Ref A non‐iterative solution of a system of ordinary differential equations arising from boundary layer theoryInternational Journal of Mathematical Education in Science and Technology, Vol. 21, No. 2 | 1 Mar 1990 Cross Ref ReferencesNonlinear Boundary Value Problems in Science and Engineering | 1 Jan 1989 Cross Ref A SOLUTION METHOD FOR A CLASS OF NONLINEAR BOUNDARY VALUE PROBLEMSChemical Engineering Communications, Vol. 43, No. 1-3 | 27 April 2007 Cross Ref Transformation of a Boundary Value Problem to an Initial Value ProblemGroup Invariance in Engineering Boundary Value Problems | 1 Jan 1985 Cross Ref Multiplicity and Stability in Distributed-Parameter Systems (DPS)Computational Methods in Bifurcation Theory and Dissipative Structures | 1 Jan 1983 Cross Ref Supplementary ReferencesGroup Analysis of Differential Equations | 1 Jan 1982 Cross Ref Optimization of nonlinear kinetic equation computationNumerical Integration of Differential Equations and Large Linear Systems | 25 August 2006 Cross Ref Gas discharges in planar low‐pressure thermionic diodes. I. Space‐charge regimeJournal of Applied Physics, Vol. 50, No. 4 | 1 Apr 1979 Cross Ref Non-linear boundary and eigenvalue problems for the emden-fowler equations by group methodsInternational Journal of Non-Linear Mechanics, Vol. 14, No. 1 | 1 Jan 1979 Cross Ref Chapter 1 IntroductionComputational Methods in Engineering Boundary Value Problems | 1 Jan 1979 Cross Ref Chapter 7 Method of Transformation—Direct TransformationComputational Methods in Engineering Boundary Value Problems | 1 Jan 1979 Cross Ref Chapter 8 Method of Transformation—Reduced Physical ParametersComputational Methods in Engineering Boundary Value Problems | 1 Jan 1979 Cross Ref Exact shooting and eigenparameter problemsNonlinear Analysis: Theory, Methods & Applications, Vol. 1, No. 1 | 1 Jan 1976 Cross Ref On the solution of Troesch's nonlinear two-point boundary value problem using an initial value methodJournal of Computational Physics, Vol. 19, No. 3 | 1 Nov 1975 Cross Ref An initial value method for the solution of the magnetohydrodynamic flow past a flat plateActa Physica Academiae Scientiarum Hungaricae, Vol. 36, No. 3 | 1 Apr 1974 Cross Ref A new method for solving eigenvalue problemsJournal of Computational Physics, Vol. 9, No. 1 | 1 Feb 1972 Cross Ref Solving Boundary-Value Problems by ImbeddingJournal of the ACM, Vol. 18, No. 4 | 1 Oct 1971 Cross Ref Transformation of boundary value problems into initial value problemsJournal of Mathematical Analysis and Applications, Vol. 32, No. 2 | 1 Nov 1970 Cross Ref Laminar boundary-layer flows of Newtonian fluids with non-Newtonian fluid injectants (Newtonian fluid laminar boundary layer flow over flat plate with nonNewtonian fluid injection)Journal of Hydronautics, Vol. 4, No. 2 | 1 Apr 1970 Cross Ref An Initial Value Method for the Solution of MHD Boundary-Layer EquationsAeronautical Quarterly, Vol. 21, No. 1 | 7 June 2016 Cross Ref An initial-value method for the solution of certain nonlinear diffusion equations in biologyMathematical Biosciences, Vol. 6 | 1 Jan 1970 Cross Ref An Initial Value Problem Approach to the Solution of Eigenvalue ProblemsSIAM Journal on Numerical Analysis, Vol. 6, No. 1 | 14 July 2006AbstractPDF (434 KB)An efficient higher-order difference method for two-dimensional structures.AIAA Journal, Vol. 7, No. 3 | 1 Mar 1969 Cross Ref High Prandtl number boundary layers with mass injection.AIAA Journal, Vol. 7, No. 3 | 1 Mar 1969 Cross Ref Drag Reduction of a Non-Newtonian Fluid by Fluid Injection at the WallJournal of Hydronautics, Vol. 2, No. 4 | 1 Oct 1968 Cross Ref 3 Examples from Transport PhenomenaNonlinear Ordinary Differential Equations in Transport Processes | 1 Jan 1968 Cross Ref Transforming Boundary Conditions to Initial Conditions for Ordinary Differential EquationsSIAM Review, Vol. 9, No. 2 | 18 July 2006AbstractPDF (486 KB)A Boundary Value ProblemSIAM Review, Vol. 7, No. 4 | 18 July 2006PDF (126 KB)Applications of differential equations in general problem solvingCommunications of the ACM, Vol. 8, No. 9 | 1 Sep 1965 Cross Ref Chapter 6 Further Approximate MethodsNonlinear Partial Differential Equations in Engineering | 1 Jan 1965 Cross Ref Chapter 2: Applications of Modern AlgebraNonlinear Partial Differential Equations in Engineering | 1 Jan 1965 Cross Ref The Finite Deflection of a Normally Loaded, Spinning, Elastic MembraneJournal of the Aerospace Sciences, Vol. 29, No. 10 | 1 Oct 1962 Cross Ref Compressive Stability of Orthotropic CylindersJournal of the Aerospace Sciences, Vol. 29, No. 10 | 1 Oct 1962 Cross Ref Volume 4, Issue 1| 1962SIAM Review1-78 History Submitted:14 August 1961Published online:18 July 2006 InformationCopyright © 1962 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1004006Article page range:pp. 43-47ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Solution of a non-linear discrete boundary value problem by a continuous analog of Newton's method

We investigate the qualitative properties of weak solutions to the boundary value problems for fourth-order linear hyperbolic equations with constant coefficients in a plane bounded domain convex with respect to characteristics. Our main scope is to prove some analog of the maximum principle, solvability, uniqueness and regularity results for weak solutions of initial and boundary value problems in the space L2. The main novelty of this paper is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions, but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L2 – data. Such solutions do not have usual traces, thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L-traces. We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. Finally, we discuss some potential challenges in applying the results and proposed methods.

The aim of this paper is to study the existence and uniqueness of solutions for a boundary value problem associated with a fractional nonlinear differential equation with higher order posed on the half-line. An appropriate continuous embedding for suitable Banach spaces are proved and the Minty–Browder theorem for monotone operators is used in the proof of existence of solutions for a boundary value problem of fractional order posed on the half-line.

Triple positive solutions of conjugate boundary value problems

Eidel’man, Butuzov, and Ivasishen. In particular, Ladyzhenskaya and Ural’tseva established sufficient conditions for the existence and uniqueness of classical and generalized solutions of main boundary-value problems in the linear case and substantiated the applicability of the Fourier method to the solution of these problems [1, pp. 295‐299]. Analogous results on the existence and uniqueness of solutions of the corresponding problems for singularly perturbed equations were obtained by Oleinik in [2]. In [3], Butuzov proposed an original method for the solution of boundary-value problems for singularly perturbed partial differential equations. According to the method of angular boundary functions developed by Butuzov, asymptotic solutions of boundary-value problems are constructed in the form of regular and boundary-layer parts. In this case, functions of the boundary-layer part are solutions of certain differential equations with constant coefficients that satisfy certain boundary conditions. In the present paper, we consider the first boundary-value problem for a degenerate, linear, singularly perturbed parabolic system. The statement of the problem is close to problems investigated by Mitropol’skii and Khoma for regularly perturbed quasilinear and nonlinear equations of the hyperbolic type [4, pp. 137‐ 225]. In the solution of the first boundary-value problem for hyperbolic equations with slowly varying coefficients, Feshchenko and Shkil’ used the Fourier method [5, pp. 226‐245]. However, the question of the convergence of the corresponding series and the possibility of their term-by-term differentiation remains open. Note that equations with slowly varying coefficients can be reduced to singularly perturbed equations by a change of the independent variable. Consider the problem