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This book comprehensively investigates various new analytical and numerical approximation techniques that are used in solving nonlinear-oscillator and structural-system problems. Students often rely on the finite element method to such an extent that on graduation they have little or no knowledge of alternative methods of solving problems. To rectify this, the book introduces several new approximation techniques.

His activities and research interests are in the field of scientific computing and numerical analysis of nonlinear parameter-dependent ordinary differential equations ODEs. He is also the founder of the Interdisciplinary Centre for Scientific Computing , where scientists of different faculties at the FSU Jena work together in the fields of applied mathematics, computer sciences and applications.

Since , he has headed an international collaborative project with the Institute of Mathematics at the National Academy of Sciences Kiev Ukraine , studying, for example, the sloshing of liquids in tanks.

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Since , Dr. In addition to his professional activities, he volunteers in various organizations and associations.

  1. Numerical methods for ordinary differential equations - Wikipedia?
  2. Statement of Problem;
  3. No. 32 in C-sharp Minor, Op. 50, No. 3!

He has also produced over 70 articles for refereed journals. His research interests include the numerical solution of ordinary differential equations ODEs , partial differential equations PDEs , integral equations, differential algebraic equations DAE and spectral methods. In addition to publishing several papers with his German colleagues, Dr. Table 8. Table 9.

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Table Fig 1. OHPM solution.

How to solve ANY differential equation

Fig 2. Fig 3. Fig 4.

Fig 5. Fig 6. Fig 7. Fig 8. Fig 9. Fig Conclusions In present work gives analytic approximate solutions for to the boundary layer flow equation over a stretching wall in presence of partial slip at the boundary.


Supporting Information. S1 Fig. S2 Fig. S3 Fig. S4 Fig. S5 Fig. S6 Fig. S7 Fig. S8 Fig. S9 Fig. S10 Fig. S11 Fig. S12 Fig. S13 Fig. S14 Fig.

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  • References 1. Sakiadis BC Boundary layer behavior on continuous solid surfaces. Boundary layer on a continuous flat surface. AIChE J. Crane LJ Flow past a stretching plate. Angew Math. Boutros YZ, A. Meccanica — View Article Google Scholar 5. Mehmood A, Ali A Analytic solution of generalized three-dimensional flow and heat transfer over a stretching plane wall.

    Mahapatra TR, Dholey S, Gupta AS Momentum and heat transfer in the magnetohydrodynamic stagnation-point flow of a viscoelastic fluid toward a stretching surface. Meccanica — View Article Google Scholar 7. Ariel PD Two-dimensional stagnation-point flow of an elasto-viscous fluid with partial slip. Meccanica — View Article Google Scholar 9.

    Pal D Heat and mass transfer in stagnation-point flow towards a stretching surface in the presence of boundary force and thermal radiation. Meccanica — View Article Google Scholar Van Gorder RA, Vajravelu K A note on flow geometries and the similarity solutions of the boundary layer equations for a nonlinearly stretching sheet. Mehmood A, Ali A Heat transfer analysis of three-dimensional flow in a channel of lower stretching wall. Nonlinear Sci. Munawar S, Mehmood A, Ali A Three-dimensional squeezing flow in a rotating channel of lower stretching porous wall.

    Mukhopadhyay S Analysis of boundary layer flow over a porous nonlinearly stretching sheet with partial slip at the boundary. Butt AS, Ali A Analysis of entropy generation effects in unsteady squeezing flow in a rotating channel with lower stretching permeable wall. Ene RD Contributions on the extension of the optimal homotopy asymptotic method in solution of the flow of the polymeric materials. He JH A coupling method of homotopy technique and perturbation technique for nonlinear problems. Non-Linear Mech. Springer Verlag, Heidelberg This "difficult behaviour" in the equation which may not necessarily be complex itself is described as stiffness , and is often caused by the presence of different time scales in the underlying problem.

    For example, a collision in a mechanical system like in an impact oscillator typically occurs at much smaller time scale than the time for the motion of objects; this discrepancy makes for very "sharp turns" in the curves of the state parameters. Stiff problems are ubiquitous in chemical kinetics , control theory , solid mechanics , weather forecasting , biology , plasma physics , and electronics.

    Analytical Approximation and Numerical Methods

    One way to overcome stiffness is to extend the notion of differential equation to that of differential inclusion , which allows for and models non-smoothness. Below is a timeline of some important developments in this field. Boundary value problems BVPs are usually solved numerically by solving an approximately equivalent matrix problem obtained by discretizing the original BVP.

    This method takes advantage of linear combinations of point values to construct finite difference coefficients that describe derivatives of the function.

    1st Edition

    For example, the second-order central difference approximation to the first derivative is given by:. One then constructs a linear system that can then be solved by standard matrix methods. For instance, suppose the equation to be solved is:. The next step would be to discretize the problem and use linear derivative approximations such as.

    On first viewing, this system of equations appears to have difficulty associated with the fact that the equation involves no terms that are not multiplied by variables, but in fact this is false. From Wikipedia, the free encyclopedia. This article includes a list of references , but its sources remain unclear because it has insufficient inline citations.

    Please help to improve this article by introducing more precise citations. April Learn how and when to remove this template message. Further information: Euler method. Further information: Backward Euler method. Further information: Exponential integrator. Main articles: Sequence , Limit mathematics , and Limit of a sequence. Further information: Truncation error numerical integration. Further information: Stiff equation. Contributions in Mathematical and Computational Sciences. Springer International Publishing. Communications of the ACM. In Bernold Fiedler ed. Mosterman ed. Model-Based Testing for Embedded Systems.

    CRC Press. Numerical methods for integration.