Numerical Methods for Differential and Integral Equations
Submission Deadline: 30 November 2019
(closed)
Guest Editors
Professor Şuayip Yüzbaşı, Department of Mathematics, Faculty of Science, Akdeniz University, Turkey
Professor Kamel Al-Khaled, Department of Mathematics, Jordan University of Science and Technology, Jordan
Professor Nurcan Baykuş Savaşaneril, Department of Mathematics, Faculty of Science, Dokuz Eylul University, Turkey
Professor Devendra Kumar, Department of Mathematics, University of Rajasthan, India
Summary
This special issue will focus on numerical solutions of diferential, integral and integro-differential equations, partial differential equations, fractional differential equations, fractional partial differential equations, stochastic partial differential equations and functional differential equations. The solutions of mentioned equations have a major role in many applied areas of science and engineering, For examle, physics, chemistry, astronomy, biology, mechanics, electronic, economics, potential theory, electrostatics. Since the mentioned equations are usually difficult to solve analytically, numerical methods are required. Therefore, this Special Issue will contribute to new numerical methods for solving the above mentioned equations and thus, many problems in science and engineering will be solved by means of new numerical methods in this special issue. We would like to invite researchers working on this topic to submit their articles to this Special Issue.
Keywords
• Numerical methods for ordinary differential equations • Numerical methods for delay differential equations • Numerical methods for partial differential equations • Numerical methods for fractional differential equations • Numerical methods for integral equations • Numerical methods for integro-differential equations • Numerical methods for model problems of differential equations • Numerical methods for fractional PDE • Numerical methods for Stochastic PDE • Numerical methods for functional differential equations
CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.3, pp. 917-939, 2020, DOI:10.32604/cmes.2020.09224
（This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)
Abstract In this paper, two classes of Riesz space fractional partial differential equations
including space-fractional and space-time-fractional ones are considered. These two
models can be regarded as the generalization of the classical wave equation in two
space dimensions. Combining with the Crank-Nicolson method in temporal direction,
efficient alternating direction implicit Galerkin finite element methods for solving these two
fractional models are developed, respectively. The corresponding stability and convergence
analysis of the numerical methods are discussed. Numerical results are provided to verify
the theoretical analysis. More >
CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.3, pp. 941-956, 2020, DOI:10.32604/cmes.2020.08938
（This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)
Abstract In this study, we present a numerical scheme similar to the Galerkin method in
order to obtain numerical solutions of the Bagley Torvik equation of fractional order 3/2. In
this approach, the approximate solution is assumed to have the form of a polynomial in the
variable t = xα
, where α is a positive real parameter of our choice. The problem is firstly
expressed in vectoral form via substituting the matrix counterparts of the terms present
in the equation. After taking inner product of this vector with nonnegative integer powers
of t up to a selected positive parameter N,… More >
CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.3, pp. 957-972, 2020, DOI:10.32604/cmes.2020.08911
（This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)
Abstract In this work, a numerical scheme is constructed for solving nonlinear parabolictype partial-integro differential equations. The proposed numerical scheme is based on
radial basis functions which are local in nature like finite difference numerical schemes.
The radial basis functions are used to approximate the derivatives involved and the integral
is approximated by equal width integration rule. The resultant differentiation matrices are
sparse in nature. After spatial approximation using RBF the partial integro-differential
equations reduce to the system of ODEs. Then ODEs system can be solved by various
types of ODE solvers. The proposed numerical scheme is tested and compared with… More >
CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.3, pp. 973-993, 2020, DOI:10.32604/cmes.2020.09329
（This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)
Abstract In this article, we approximate the solution of high order linear Fredholm
integro-differential equations with a variable coefficient under the initial-boundary
conditions by Bell polynomials. Using collocation points and treating the solution as a
linear combination of Bell polynomials, the problem is reduced to linear system of
equations whose unknown variables are Bell coefficients. The solution to this algebraic
system determines the approximate solution. Error estimation of approximate solution is
done. Some examples are provided to illustrate the performance of the method. The numerical
results are compared with the collocation method based on Legendre polynomials and the
other two methods… More >
CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.2, pp. 739-776, 2020, DOI:10.32604/cmes.2020.08563
（This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)
Abstract Based on rectangular partition and bilinear interpolation, we construct an
alternating-direction implicit (ADI) finite volume element method, which combined the
merits of finite volume element method and alternating direction implicit method to solve
a viscous wave equation with variable coefficients. This paper presents a general procedure
to construct the alternating-direction implicit finite volume element method and gives
computational schemes. Optimal error estimate in L2 norm is obtained for the schemes.
Compared with the finite volume element method of the same convergence order, our
method is more effective in terms of running time with the increasing of the computing
scale. Numerical… More >
Muhammad Altaf Khan, Khanadan Khan, Mohammad A. Safi, Mahmoud H. DarAssi
CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.2, pp. 777-795, 2020, DOI:10.32604/cmes.2020.08208
（This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)
Abstract The present paper investigates the theoretical analysis of the tuberculosis (TB) model in the discrete-time case. The model is parameterized by the TB
infection cases in the Pakistani province of Khyber Pakhtunkhwa between 2002
and 2017. The model is parameterized and the basic reproduction number is
obtained and it is found R0 ¼ 1:5853. The stability analysis for the model is presented and it is shown that the discrete-time tuberculosis model is stable at the
disease-free equilibrium whenever R0 < 1 and further we establish the results
for the endemic equilibria and prove that the model is globally asymptotically
stable… More >
CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.2, pp. 797-819, 2020, DOI:10.32604/cmes.2020.08717
（This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)
Abstract In this paper, we approximate the solution and also discuss the periodic behavior
termed as eventual periodicity of solutions of (IBVPs) for some dispersive wave equations
on a bounded domain corresponding to periodic forcing. The constructed numerical
scheme is based on radial kernels and local in nature like finite difference method. The temporal variable is executed through RK4 scheme. Due to the local nature and sparse differentiation matrices our numerical scheme efficiently recovers the solution. The results
achieved are validated and examined with other methods accessible in the literature. More >
CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.1, pp. 333-351, 2020, DOI:10.32604/cmes.2020.08097
（This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)
Abstract Wavelength-dependent mathematical modelling of the differential energy change
of a photon has been performed inside a proposed hypothetical optical medium. The existence of this medium demands certain mathematical constraints, which have been derived
in detail. Using reverse modelling, a medium satisfying the derived conditions is proven to
store energy as the photon propagates from the entry to exit point. A single photon with a
given intensity is considered in the analysis and hypothesized to possess a definite non-zero
probability of maintaining its energy and velocity functions analytic inside the proposed
optical medium, despite scattering, absorption, fluorescence, heat generation, and other… More >
CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.1, pp. 353-376, 2020, DOI:10.32604/cmes.2020.08776
（This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)
Abstract In this paper, a special case of nonlinear time fractional cable equation is studied.
For the equation defined on a bounded domain, the existence, uniqueness, and regularity
of the solution are firstly studied. Furthermore, it is numerically studied via the weighted
and shifted Grünwald difference (WSGD) methods/the local discontinuous Galerkin (LDG)
finite element methods. The derived numerical scheme has been proved to be stable and
convergent with order O(∆t2 + hk+1), where ∆t, h, k are the time stepsize, the spatial
stepsize, and the degree of piecewise polynomials, respectively. Finally, a numerical
experiment is presented to verify the theoretical analysis. More >
Muhammad Shuaib, Muhammad Bilal, Muhammad Altaf Khan, Sharaf J. Malebary
CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.1, pp. 377-400, 2020, DOI:10.32604/cmes.2020.08076
（This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)
Abstract An unsteady viscous fluid flow with Dufour and Soret effect, which results
in heat and mass transfer due to upward and downward motion of flexible rotating disk,
has been studied. The upward or downward motion of non rotating disk results in
two dimensional flow, while the vertical action and rotation of the disk results in three
dimensional flow. By using an appropriate transformation the governing equations are
transformed into the system of ordinary differential equations. The system of ordinary
differential equations is further converted into first order differential equation by selecting
suitable variables. Then, we generalize the model by using… More >
Yanlong Zhang, Baoli Yin, Yue Cao, Yang Liu, Hong Li
CMES-Computer Modeling in Engineering & Sciences, Vol.122, No.3, pp. 1081-1098, 2020, DOI:10.32604/cmes.2020.07822
（This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)
Abstract In this article, a high-order scheme, which is formulated by combining the
quadratic finite element method in space with a second-order time discrete scheme, is
developed for looking for the numerical solution of a two-dimensional nonlinear time
fractional thermal diffusion model. The time Caputo fractional derivative is approximated
by using the L2 -1 σ formula, the first-order derivative and nonlinear term are discretized
by some second-order approximation formulas, and the quadratic finite element is used to
approximate the spatial direction. The error accuracy O(h3 + ∆t2 ) is obtained, which is
verified by the numerical results. More >
Naeem Faraz, Yasir Khan, Amna Anjum, Anwar Hussain
CMES-Computer Modeling in Engineering & Sciences, Vol.122, No.3, pp. 1099-1118, 2020, DOI:10.32604/cmes.2020.08389
（This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)
Abstract Current research is about the injection of a viscous fluid in the presence of a
transverse uniform magnetic field to reduce the sliding drag. There is a slip-on both the
slider and the ground in the two cases, for example, a long porous slider and a circular
porous slider. By utilizing similarity transformation Navier-Stokes equations are
converted into coupled equations which are tackled by Integral Transform Method.
Solutions are obtained for different values of Reynolds numbers, velocity slip, and
magnetic field. We found that surface slip and Reynolds number has a substantial
influence on the lift and drag of long… More >
Abdulghani Alharbi, Mahmoud A. E. Abdelrahman, M. B. Almatrafi
CMES-Computer Modeling in Engineering & Sciences, Vol.122, No.2, pp. 743-756, 2020, DOI:10.32604/cmes.2020.07996
（This article belongs to this Special Issue: Numerical Methods for Differential and Integral Equations)
Abstract The nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM)
equation is solved numerically using adaptive moving mesh PDEs (MMPDEs) method.
Indeed, the exact solution of the DMBBM equation is obtained by using the extended
Jacobian elliptic function expansion method. The current methods give a wider
applicability for handling nonlinear wave equations in engineering and mathematical
physics. The adaptive moving mesh method is compared with exact solution by numerical
examples, where the explicit solutions are known. The numerical results verify the accuracy
of the proposed method. More >
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