scholarly journals An Exact Analytical Solution to the Shallow Water Equations Near Beaches

2017 ◽  
Vol 71 ◽  
pp. 63
Author(s):  
M. Yourdkhani ◽  
Saber Zarrinkamar
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Shallow Water Equations ◽  
Water Waves ◽  
Jacobi Polynomials ◽  
Exact Analytical Solution ◽  
Second Order ◽  
Shallow Waters ◽  
Order Polynomial ◽  
Linear Counterpart

<p>The nonlinear differential equation governing the dynamics of water waves can be well approximated by a linear counterpart in the case of shallow waters near beaches. The linear equation, which is of second order nature, cannot be exactly solved in many apparently simple cases. In our work, we consider the shape of system as a complete second-order polynomial which contains the constant (step-like), linear and quadratic shapes near the beach. We then apply some novel transformations and transform the problem into a form which can be solved in an exact analytical manner via the powerful Nikiforov-Uvarov technique. The eigenfunctions of the problem are obtained in terms of the Jacobi polynomials and the eigenvalue equation is reported for any arbitrary mode. </p>

scholarly journals Numerical Solution of the Bagley-Torvik Equation Using the Integer-Order Derivatives Expansion

2018 ◽  
Vol 6 (2) ◽  
Author(s):  
Afrah Sadiq Hasan
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Numerical Solution ◽  
Fractional Order ◽  
Infinite Series ◽  
Exact Analytical Solution ◽  
Second Order ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Integer Order

Numerical solution of the well-known Bagley-Torvik equation is considered. The fractional-order derivative in the equation is converted, approximately, to ordinary-order derivatives up to second order. Approximated Bagley-Torvik equation is obtained using finite number of terms from the infinite series of integer-order derivatives expansion for the Riemann–Liouville fractional derivative. The Bagley-Torvik equation is a second-order differential equation with constant coefficients. The derived equation, by considering only the first three terms from the infinite series to become a second-order ordinary differential equation with variable coefficients, is numerically solved after it is transformed into a system of first-order ordinary differential equations. The approximation of fractional-order derivative and the order of the truncated error are illustrated through some examples. Comparison between our result and exact analytical solution are made by considering an example with known analytical solution to show the preciseness of our proposed approach.

scholarly journals An analytical approximation of the transient response of a voltage dependent supercapacitor model

2019 ◽  
Vol 39 (3) ◽  
pp. 310-319
Author(s):  
Tomislav Barić ◽  
Hrvoje Glavaš ◽  
Ružica Kljajić
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Nonlinear Differential Equation ◽  
Exact Analytical Solution ◽  
Initial Response ◽  
Analytical Approximation ◽  
Weight Functions ◽  
Transient Phenomenon ◽  
Voltage Dependent ◽  
Analytical Expressions

Supercapacitors are well known for their voltage dependent capacity. Due to this, it is not possible to obtain the exact analytical solution of the nonlinear differential equation which describes the transient charging and discharging. For this reason, approximations of differential equations must be carried out in order to obtain an approximate analytical solution. The focus of this paper is on a different approach. Instead of approximating the differential equation and obtaining analytical expressions for such approximations, an intuitive approach is chosen. This approach is based on the separation of the initial response from the rest of the transient phenomenon. Both parts of the transient phenomenon are described with adequate functions. Using appropriate weight functions, both functions are combined into a single function that describes the whole transient phenomenon. As shown in the paper, such an approach gives an excellent description of the whole transient. Also, it provides simpler expressions compared to those obtained by approximation of the nonlinear differential equation. With respect to their accuracy, these expressions do not lag behind the aforementioned approach. The validity of the presented analytical expressions was confirmed by comparing their results with those obtained by numerically solving the nonlinear differential equation.

Notes on the Riccati Equation

2005 ◽  
Vol 70 (7) ◽  
pp. 941-950 ◽  
Author(s):  
Eugene S. Kryachko
Keyword(s):  
Differential Equation ◽  
Normal Form ◽  
Riccati Equation ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Riccati Transformation ◽  
Rational Solutions ◽  
Order Polynomial ◽  
Linear Differential ◽  
The Relationship

The relationship between the Riccati and Schrödinger equations is discussed. It is shown that the transformation converting the Riccati equation into its normal form is expressed in terms of the roots of its algebraic part treated as a second-order polynomial. Together with the well-known Riccati transformation, a new transformation which also links the Riccati equation to the second-order linear differential equation is introduced. The latter is actually the Riccati transformation applied to an "inverse" Riccati equation. Two specific forms of the Riccati equation admitting the explicit particular rational solutions are obtained.

scholarly journals Modified Kudrayshov Method to Solve Generalized Kuramoto–Sivashinsky Equation

2018 ◽  
Author(s):  
Adem Kilicman ◽  
Rathinavel Silambarasan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Exact Solutions ◽  
Exact Analytical Solution ◽  
Nonlinear Partial Differential Equation ◽  
Complex Structures ◽  
Algebraic Equations ◽  
Uniqueness Of Solutions ◽  
Sivashinsky Equation

The generalized Kuramoto&ndash;Sivashinsky equation is investigated using the modified Kudrayshov method for the exact analytical solution. The modified Kudrayshov method converts the nonlinear partial differential equation to algebraic equations, as a result of various steps, which on solving the so obtained equation systems yields the analytical solution. By this way various exact solutions including complex structures are found and drawn their behaviour in complex plane by Maple to compare the uniqueness of solutions.

Load-Supporting Fluid-Filled Cylindrical Membranes

1985 ◽  
Vol 52 (4) ◽  
pp. 913-918 ◽  
Author(s):  
V. Namias
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Nonlinear Differential Equation ◽  
Entire Range ◽  
Exact Analytical Solution ◽  
Asymptotic Expressions ◽  
Flexible Membranes ◽  
Approximate Expressions

When long cylindrical flexible membranes are filled with a fluid and used to support external weights, the shape they assume and the relevant geometrical and dynamical quantities are governed by a nonlinear differential equation subject to particular boundary conditions. First, a complete and exact analytical solution is obtained for an unloaded membrane. Very accurate approximate expressions are derived directly from the exact solution for the entire range of applied pressures and fluid densities. Next, the nonlinear differential equation is solved exactly under boundary conditions corresponding to the loading of the membrane. Simple asymptotic expressions are also obtained in the limit of large loads.

scholarly journals Modified Kudrayshov Method to Solve Generalized Kuramoto–Sivashinsky Equation

2018 ◽  
Author(s):  
Rathinavel Silambarasan ◽  
Adem Kilicman
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Complex Plane ◽  
Exact Analytical Solution ◽  
Nonlinear Partial Differential Equation ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Complex Solutions ◽  
Sivashinsky Equation

The generalized Kuramoto&ndash;Sivashinsky equation is investigated using the modified Kudrayshov equation for the exact analytical solution. The modified Kudrayshov method converts the nonlinear partial differential equation to algebraic equations by results of various steps which on solving the so obtained equation systems yields the analytical solution. By this way various exact including complex solutions are found and drawn their behaviour in complex plane by Maple to compare the uniqueness of various solutions.

The Shakedown Boundary for Parabolic Stress Distribution in NB-3222.5

2013 ◽  
Author(s):  
W. Reinhardt ◽  
A. Asadkarami
Keyword(s):  
Thermal Stress ◽  
Temperature Distribution ◽  
Stress Distribution ◽  
Analytical Solution ◽  
Thermal Loading ◽  
Exact Analytical Solution ◽  
Second Order ◽  
Fe Model ◽  
Membrane Stress ◽  
Shakedown Analysis

The rules for the prevention of thermal stress ratchet in NB-3222.5 address the interaction of general primary membrane stress with two types of cyclic thermal loading. The first is a linear through-wall temperature gradient, for which the shakedown boundary is given by the well-known Bree diagram. The Code provides a second shakedown boundary for the interaction of general primary membrane stress with a “parabolic” temperature distribution. The corresponding ratchet boundary is fully defined in the elastic range, but only three points are given in the elastic-plastic regime. The range of validity of this ratchet boundary in terms of the thermal stress distribution (does “parabolic” mean second-order in the thickness coordinate or any polynomial of degree greater than one? If it is second-order, are there any further restrictions?) is not well defined in NB-3222.5. Using a direct lower bound method of shakedown analysis, the non-cyclic method, an exact analytical solution is derived for the shakedown boundary corresponding to the interaction of general primary membrane stress with a cyclic “parabolic” temperature distribution. By comparison to what is given in NB-3222.5, the thermal condition for which the Code equation is valid is defined and its range of validity is established. To study the transition behavior to the steady state and to confirm the analytical solution, numerical results using an FE model are also obtained.

Second-order isothermal reaction in a semi-batch reactor: modeling, exact analytical solution, and experimental verification

2019 ◽  
Vol 4 (11) ◽  
pp. 2011-2020
Author(s):  
Mana Kord ◽  
Ali Nematollahzadeh ◽  
Behruz Mirzayi
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Flow Rate ◽  
Experimental Verification ◽  
Exact Analytical Solution ◽  
Batch Reactor ◽  
Second Order ◽  
Reaction Conditions ◽  
Reactor Modeling ◽  
Isothermal Reaction

Mathematical model of a semi-batch reactor (SBR) can be employed for tuning the concentration or flow rate of the external-feed of reactants, to control the reaction conditions and product properties.

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