The Influence of Non-Isothermal Conditions on Pressure Head Jumps in the Nonlinear Consolidation Problem in the Presence of Geobarriers

Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions

One parameter operator imbedding to modify Newton method for solution of nonlinear equations

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

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 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. 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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

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.

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.

A method is described for constructing mathematical models of interrelated processes in porous media that are complex multicomponent systems. The performance capabilities of the package FreeFem++ are shown as applied to solving corresponding free boundary-value problems for systems of quasilinear parabolic equations by the finite element method using the parallelization of computations.

The character of the proper refinement of the elements (mesh) around the boundary is studied. It is shown that it is possible to obtain the highest (optimal) rate of convergence by this refinement.

The problem of loss of stability and post-critical behavior of a compressed orthotropic rectangular plate on a nonlinearly elastic base containing continuously distributed fields of dislocations and disclinations and under the influence of a small normal load is considered. The compressive force components are evenly distributed along the edges and act parallel to the main directions of elasticity. The problem is formulated as an analogue of a system of nonlinear Karman equations for an orthotropic plate containing a function called the incompatibility measure, which is expressed in terms of the densities of edge dislocations and wedge disclinations. The system of equations takes into account the small transverse pressure and the reaction of the elastic base in the form of a polynomial of the third degree of deflection. The following boundary conditions are considered: the edges of the plate are freely pinched or movably pivotally supported; two opposite edges of the plate are freely pinched or movably pivotally supported, and the other two are free from loads. The stress function is sought in the form of two components: the stress function caused by the presence of internal sources, determined from the linear boundary value problem, and the stress function caused by the external impact of compressive loads and a nonlinear elastic base, which is determined from the nonlinear boundary value problem. The nonlinear boundary value problem is investigated by the Lyapunov – Schmidt method. To solve the linearized equation from which the critical value of the compressive load is determined, the variational method is used in combination with the difference method. A system of branching equations of the Lyapunov – Schmidt method is constructed, which is investigated by numerical methods. The post-critical behavior of the plate is investigated and asymptotic formulas for new equilibria in the vicinity of the critical load are derived. For various values of compressive loads, the orthotropy parameters of the plate and the intensity parameter of internal stresses, the relations between the values of the base parameters are established, at which its bearing capacity is preserved in the vicinity of the classical value of the critical load.

A new highly accurate algorithm for the solution of nonlinear singular boundary value problems for ordinary differential equations is presented.The algorithm uses a collocation technique based on polynomial approximation at Sinc points. The scheme is tested for some nonlinear singular boundaryvalue problems showing an exponential convergence rate. The examples areof second and higher order singular, nonlinear boundary value problems. Foreach example the error formula of the approximation is discussed and verifiledin a comparison of the analytic solution

A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems

This paper considers a free boundary problem for a system of quasi-linear parabolic equations in one dimension. Nonlinear problems with a free boundary are studied using a method based on constructing a priori estimates. For the solutions of the problem, apriory estimates of Shauder type are established. On the base of apriory estimates, the existence and uniqueness theorems are proved.

The article proposes original solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem. And we use the matrix pseudo-inversion technique of Moore-Penrose. The posed problem in the article continues the study of conditions of solvability and schemes for finding solutions of the nonlinear Noetherian boundary-value problems given in the monographs by A. Poincare, A.M. Lyapunov, I.G. Malkin, J. Hale, Yu.A. Ryabov, A.M. Samoylenko, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina and A.A. Boychuk. We studied a general case, when a linear bounded operator corresponding to the homogeneous part of the linear Noetherian differential-algebraic boundary value problem has no inverse. Sufficient conditions for reducibility of the differential algebraic equation to the system uniting a differential and algebraic equation are found. Thus, the differential-algebraic boundary value problem is reduced to the nonlinear Noetherian boundary value problem for the system of ordinary differential equations. We studied the case of the presence of simple roots of the equation for generating amplitudes. Constructive necessary and sufficient conditions of existence were obtained to find solutions to the problem in the critical case, and the converging iterative scheme was constructed. The proposed solvability conditions, and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem are illustrated in detail by the example from the nonlinear Noetherian differential-algebraic boundary value problem for Duffing type equations. For control of the rate of the iterative scheme convergence to the exact solution of the differential-algebraic boundary value problem for the Duffing type equation, we used the residuals of the obtained approximations in the Duffing type equation in the space of continuous functions.

Soil environments are heterogeneous in their nature. This heterogeneity creates significant difficulties both in terms of construction practice and in terms of the mathematical modeling and computer simulation of the physical-chemical processes in these heterogeneous soil arrays. From the standpoint of mathematical modeling, the issue is the discontinuity of functions, which characterize the examined processes, on such inclusions. Moreover, the characteristics of such inclusions may depend on the defining functions of the processes studied (head, temperature, humidity, the concentration of chemicals, and their gradients). And this requires the modification of conjugation conditions and leads to the nonlinear boundary-value problems in heterogeneous areas. That is why this work has examined the impact of the existence of thin inclusions on the conjugation conditions for the defining functions of the filtration and geomigration processes on them. The conjugation condition for heads has also been modified while the mathematical model of an elastic filtration mode in a heterogeneous array of soil, which contains thin weakly permeable inclusions, has been improved. The improvement implies the modification of conjugation conditions for heads on thin inclusions when the filtering factor of the inclusion itself is nonlinearly dependent on the head gradient. The numerical solution to the corresponding nonlinear boundary-value problem has been found using a finite-element method. A series of numerical experiments were conducted and their analysis was carried out. The possibility of a significant impact on the head jump has been shown taking into consideration the dependence of filtration characteristics of an inclusion on head gradients. In particular, the relative difference of head jumps lies between 26 % and 99 % relative to the problem with a stable filtration factor for an inclusion. In other words, when conducting forecast calculations, the influence of such dependences cannot be neglected