Radial symmetry for an elliptic PDE with a free boundary

In this paper we prove symmetry for solutions to the semi-linear elliptic equation Δ u = f ( u ) in B 1 , 0 ≤ u > M , in B 1 , u = M , on ∂ B 1 , \begin{equation*} \Delta u = f(u) \quad \text { in } B_1, \qquad 0 \leq u > M, \quad \text { in } B_1, \qquad u = M, \quad \text { on } \partial B_1, \end{equation*} where M > 0 M>0 is a constant, and B 1 B_1 is the unit ball. Under certain assumptions on the r.h.s. f ( u ) f (u) , the C 1 C^1 -regularity of the free boundary ∂ { u > 0 } \partial \{u>0\} and a second order asymptotic expansion for u u at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin’s celebrated boundary point lemma, which is not available in our case due to lack of C 2 C^2 -regularity of solutions.

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Differential quadrature method for nonlinear fractional partial differential equations

Engineering Computations ◽

10.1108/ec-04-2018-0179 ◽

2018 ◽

Vol 35 (6) ◽

pp. 2349-2366 ◽

Cited By ~ 3

Author(s):

Umer Saeed ◽

Mujeeb ur Rehman ◽

Qamar Din

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

High Efficiency ◽

Differential Quadrature Method ◽

Differential Quadrature ◽

Operational Matrices ◽

Fractional Partial Differential Equations ◽

Infinite Domain ◽

Partial Differential ◽

Content Type

Purpose The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more accurate results. Design/methodology/approach The authors proposed a method by using the Chebyshev wavelets in conjunction with differential quadrature technique. The operational matrices for the method are derived, constructed and used for the solution of nonlinear fractional partial differential equations. Findings The operational matrices contain many zero entries, which lead to the high efficiency of the method and reasonable accuracy is achieved even with less number of grid points. The results are in good agreement with exact solutions and more accurate as compared to Haar wavelet method. Originality/value Many engineers can use the presented method for solving their nonlinear fractional models.

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An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

International Journal of Numerical Methods for Heat &amp Fluid Flow ◽

10.1108/hff-12-2017-0511 ◽

2018 ◽

Vol 28 (11) ◽

pp. 2620-2649 ◽

Cited By ~ 2

Author(s):

Rajni Rohila ◽

R.C. Mittal

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Numerical Solutions ◽

Diffusion Equations ◽

Spline Functions ◽

Two Dimensional ◽

Partial Differential ◽

Content Type ◽

B Spline ◽

Advection Diffusion

Purpose This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method. Design/methodology/approach A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids. Findings Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also. Originality/value ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

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Thermal breakdown in high voltage direct current cable insulations due to space charges

COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering ◽

10.1108/compel-12-2017-0531 ◽

2018 ◽

Vol 37 (5) ◽

pp. 1689-1697 ◽

Cited By ~ 2

Author(s):

Christoph Jörgens ◽

Markus Clemens

Keyword(s):

Electric Field ◽

Breakdown Voltage ◽

High Voltage ◽

Direct Current ◽

Aging Effect ◽

Radial Symmetry ◽

Thermal Breakdown ◽

Content Type ◽

High Voltage Direct Current ◽

Space Charges

Purpose In high voltage direct current (HVDC), power cables heat is generated inside the conductor and the insulation during operation. A higher amount of the generated heat in comparison to the dissipated one, results in a possible thermal breakdown. The accumulation of space charges inside the insulation results in an electric field that contributes to the geometric electric field, which comes from the applied voltage. The total electric field decreases in the vicinity of the conductor, while it increases near the sheath, causing a possible change of the breakdown voltage. Design/methodology/approach Here, the thermal breakdown is studied, also incorporating the presence of space charges. For a developed electro-thermal HVDC cable model, at different temperatures, the breakdown voltage is computed through numerical simulations. Findings The simulation results show a dependence of the breakdown voltage on the temperature at the location of the sheath. The results also show only limited influence of the space charges on the breakdown voltage. Research limitations/implications The study is restricted to one-dimensional problems, using radial symmetry of the cable, and does not include any aging or long-term effect of space charges. Such aging effect can locally increase the electric field, resulting in a reduced breakdown voltage. Originality/value A comparison of the breakdown voltage with and without space charges is novel. The chosen approach allows for the first time to assess the influence of space charges and field inversion on the thermal breakdown.

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Reconstruction of the parameter in parabolic partial differential equations using Haar wavelet method

Engineering Computations ◽

10.1108/ec-03-2020-0163 ◽

2020 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Gopal Priyadarshi ◽

B.V. Rathish Kumar

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Computational Cost ◽

Haar Wavelet ◽

Haar Wavelets ◽

Parabolic Partial Differential Equations ◽

Partial Differential ◽

Wavelet Method ◽

Content Type ◽

Haar Wavelet Method

Purpose In the past few years, Haar wavelet-based numerical methods have been applied successfully to solve linear and nonlinear partial differential equations. This study aims to propose a wavelet collocation method based on Haar wavelets to identify a parameter in parabolic partial differential equations (PDEs). As Haar wavelet is defined in a very simple way, implementation of the Haar wavelet method becomes easier than the other numerical methods such as finite element method and spectral method. The computational time taken by this method is very less because Haar matrices and Haar integral matrices are stored once and used for each iteration. In the case of Haar wavelet method, Dirichlet boundary conditions are incorporated automatically. Apart from this property, Haar wavelets are compactly supported orthonormal functions. These properties lead to a huge reduction in the computational cost of the method. Design/methodology/approach The aim of this paper is to reconstruct the source control parameter arises in quasilinear parabolic partial differential equation using Haar wavelet-based numerical method. Haar wavelets possess various properties, for example, compact support, orthonormality and closed form expression. The main difficulty with the Haar wavelet is its discontinuity. Therefore, this paper cannot directly use the Haar wavelet to solve partial differential equations. To handle this difficulty, this paper represents the highest-order derivative in terms of Haar wavelet series and using successive integration this study obtains the required term appearing in the problem. Taylor series expansion is used to obtain the second-order partial derivatives at collocation points. Findings An efficient and accurate numerical method based on Haar wavelet has been proposed for parameter identification in quasilinear parabolic partial differential equations. Numerical results are obtained from the proposed method and compared with the existing results obtained from various finite difference methods including Saulyev method. It is shown that the proposed method is superior than the conventional finite difference methods including Saulyev method in terms of accuracy and CPU time. Convergence analysis is presented to show the accuracy of the proposed method. An efficient algorithm is proposed to find the wavelet coefficients at target time. Originality/value The outcome of the paper would have a valuable role in the scientific community for several reasons. In the current scenario, the parabolic inverse problem has emerged as very important problem because of its application in many diverse fields such as tomography, chemical diffusion, thermoelectricity and control theory. In this paper, higher-order derivative is represented in terms of Haar wavelet series. In other words, we represent the solution in multiscale framework. This would enable us to understand the solution at various resolution levels. In the case of Haar wavelet, this paper can achieve a very good accuracy at very less resolution levels, which ultimately leads to huge reduction in the computational cost.

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A numerical algorithm based on scale-3 Haar wavelets for fractional advection dispersion equation

Engineering Computations ◽

10.1108/ec-01-2020-0013 ◽

2020 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Sapna Pandit ◽

R.C. Mittal

Keyword(s):

Haar Wavelet ◽

Haar Wavelets ◽

System Control ◽

Partial Differential ◽

Content Type ◽

Advection Dispersion ◽

Novel Approach ◽

Gauss Elimination ◽

Time Variables ◽

First Time

Purpose This paper aims to propose a novel approach based on uniform scale-3 Haar wavelets for unsteady state space fractional advection-dispersion partial differential equation which arises in complex network, fluid dynamics in porous media, biology, chemistry and biochemistry, electrode – electrolyte polarization, finance, system control, etc. Design/methodology/approach Scale-3 Haar wavelets are used to approximate the space and time variables. Scale-3 Haar wavelets converts the problems into linear system. After that Gauss elimination is used to find the wavelet coefficients. Findings A novel algorithm based on Haar wavelet for two-dimensional fractional partial differential equations is established. Error estimation has been derived by use of property of compactly supported orthonormality. The correctness and effectiveness of the theoretical arguments by numerical tests are confirmed. Originality/value Scale-3 Haar wavelets are used first time for these types of problems. Second, error analysis in new work in this direction.

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Boundedness and regularity of solutions of degenerate elliptic partial differential equations

Journal of Differential Equations ◽

10.1016/j.jde.2017.05.001 ◽

2017 ◽

Vol 263 (7) ◽

pp. 3714-3736 ◽

Cited By ~ 2

Author(s):

Seng-Kee Chua

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Elliptic Partial Differential Equations ◽

Regularity Of Solutions ◽

Partial Differential ◽

Degenerate Elliptic

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Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm

International Journal of Numerical Methods for Heat &amp Fluid Flow ◽

10.1108/hff-07-2016-0278 ◽

2018 ◽

Vol 28 (4) ◽

pp. 828-856 ◽

Cited By ~ 61

Author(s):

Omar Abu Arqub

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Exact Solutions ◽

Reproducing Kernel ◽

Numerical Solutions ◽

Fluid Flows ◽

Kernel Functions ◽

Fractional Partial Differential Equations ◽

Partial Differential ◽

Content Type

Purpose The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit. Design/methodology/approach The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions. Findings Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models. Research limitations/implications Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers. Practical implications The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability. Social implications Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest. Originality/value This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.

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Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity

Journal of Mathematical Physics ◽

10.1063/1.5107517 ◽

2019 ◽

Vol 60 (10) ◽

pp. 101509 ◽

Cited By ~ 5

Author(s):

Kamel Saoudi ◽

Sekhar Ghosh ◽

Debajyoti Choudhuri

Keyword(s):

Regularity Of Solutions ◽

Hölder Regularity ◽

Elliptic Pde ◽

Holder Regularity

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Optimization of coal handling system performability for a thermal power plant using PSO algorithm

Grey Systems Theory and Application ◽

10.1108/gs-01-2020-0002 ◽

2020 ◽

Vol 10 (3) ◽

pp. 359-376

Author(s):

Subhash Malik ◽

P.C. Tewari

Keyword(s):

Partial Differential Equations ◽

Power Plant ◽

Differential Equations ◽

Thermal Power Plant ◽

Thermal Power ◽

Performance Model ◽

Transition Diagram ◽

Partial Differential ◽

Content Type ◽

Handling System

PurposeThis paper deals with the optimization of coal handling system performability for a thermal power plant.Design/methodology/approachCoal handling system comprises of five subsystems, namely Wagon Tippler, Crusher, Bunker, Feeder and Coal Mill. The partial differential equations are derived on the behalf of transition diagram by using the Markov approach. These partial differential equations are further solved to obtain the performance model with the help of normalization condition. Numerous performability levels are achieved by putting the appropriate combinations of failure and repair rates (FRRs) in performance model. Performability optimization for coal handling system is obtained by varying the population and generation size.FindingsHighest performability level, that is, 93.33 at population size of 40 and 93.31 at generation size of 70, is observed.Originality/valueThe findings of this paper highlight the optimum value of performability level and FRRs for numerous subsystems. These findings are highly beneficial for plant administration to decide about the maintenance planning.

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