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In this paper, we propose an improved Generalize Finite Integration Method with Volterra operator to solve multi-dimensional biharmonic equations. The novelty of the proposed method is that its mth order integration matrix will be exactly formulated by using Volterra operator. This leads to a significant improvement in the accuracy of the approximation. Furthermore, the original strict requirement of uniformly distributed collocation points is waived in the proposed method due to the use of piecewise polynomials. This contributes to another distinct advantage in placing collocation points for problems with geometrically complex domains. Four numerical examples are constructed to demonstrate these advantages. Comparisons on the accuracies obtained from quadrature formulas with and without Volterra operator are given. Numerical results indicate that the proposed method is capable for solving multi-dimensional biharmonic equations with superior efficiency, higher accuracy and fewer memory usage.

In this paper, we combine the recently developed localized radial basis functions-based pseudo-spectral method with the time-splitting technique to solve a linear wave equation arising from modelling the wave dynamics using peridynamic formulation in continuum mechanics. Specifically, we adopt this combined method for solving a Hamiltonian ordinary differential equation system, which is equivalent to the original linear peridynamic equation after introducing a new variable. The proposed approach inherits advantages of these two related methods in space and time: (1) offering high accuracy and efficiency in the solution of the problem under irregular domains for both uniform and non-uniform discretizations; (2) extending the applicability of the approach to multi-dimensions; and (3) maintaining a good approximation for problems at large time-step and long time integration. Numerical results indicate that the proposed method is simple, accurate, efficient, and stable for solving various linear peridynamic problems.

In this paper, we generalize the recently developed Finite Integration Method (FIM) for the solutions of high-dimensional partial differential equations. Formulation of this Generalized Finite Integration Method (GFIM) can be derived due to the use of piecewise polynomials in the numerical integrations. The GFIM does not require the strict requirement for uniformly distributed nodal points in the original FIM. This robustness advantage extends the applicability of FIM to solve partial differential equations by using direct Kronecker product. Due to the unconditional stability of numerical integrations, the GFIM is effective and efficient to solve higher dimensional partial differential equations with stiffness. For numerical verification, we construct several 1D to 4D problems with different types of stiffness and make comparisons among existing numerical methods.

In this paper we consider the two-dimensional nonlocal phase-field and hyperbolic nonlocal phase-field models to obtain their numerical solutions. For this purpose, we propose a localized method based on radial basis functions (RBFs), namely localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for spatial discretization. The basic idea of the LRBF-PSM is to construct a set of orthogonal functions by RBFs on each overlapping sub-domain from which the global solution can be obtained by extending the approximation on each sub-domain to the entire domain. This approach does not require meshing in spatial domain and hence inherits the meshless and spectral convergence properties of the global radial basis functions collocation method (GRBFCM). Some numerical results indicate that the obtained simulations via the LRBF-PSM is effective and stable for approximating the solution of nonlocal models investigated in the current paper.

Spectral/pseudo-spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations. In this paper, we will present the use of a localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for solving 2D nonlocal problems with radial nonlocal kernels. The basic idea of the LRBF-PSM is to construct a set of orthogonal functions by RBFs on each overlapping sub-domain from which the global solution can be obtained by extending the approximation on each sub-domain to the entire domain. Numerical implementation indicates that the proposed LRBF-PSM is simple to use, efficient and robust to solve various nonlocal problems.

We combine in this paper the recently developed finite integration method (FIM) with radial basis function (FIM–RBF) to solve stiff problems. The idea of FIM is to transform partial differential equations (PDEs) into integral equations whose approximations can be stably and accurately obtained from standard numerical quadratures. This contributes to one distinct benefit: smoothing out the instability of solution process in solving stiff problems modelled by PDEs. The other distinct benefit comes from the truly meshfree advantage of radial basis function (RBF) method for solving various kinds of well-posed PDEs with superior accuracy. Since the RBF method gives an intrinsic full, and hence ill-conditioned, resultant matrix, it fails to tackle stiff problems. The combination of FIM and RBF method enjoys not only the distinct benefit of smoothing out stiffness but also gives a much less ill-conditioned resultant matrix. As a result, the headache problem on choosing critical value of shape parameter for spectral convergence in RBF does not persist. Numerical results indicated that the accuracy of the FIM–RBF has been increased by approximately two order of magnitude with only one third of CPU time in comparing to some other spectral methods such as wavelet adaptive scheme.

This paper investigates the temporal effects in the modeling of flows through porous media and particles transport. Studies will be made among the time fractional diffusion model and two classical nonlinear diffusion models. The effects of the parameters upon the mentioned models have been studied. By simulating the sub-diffusion processes and comparing the numerical results of these models under different boundary conditions, we can conclude that the time fractional diffusion model is more suitable for simulating the sub-diffusion with steady diffusion rate; whereas the nonlinear models are more appropriate for depicting the sub-diffusion under changing diffusion rate.

In this paper, the localized radial basis function collocation method (LRBFCM) is combined with the partial upwind scheme for solving convection-dominated fluid flow problems. The localization technique adopted in LRBFCM has shown to be effective in avoiding the well known ill-conditioning problem of traditional meshless collocation method with globally defined radial basis functions (RBFs). For convection–diffusion problems with dominated convection, stiffness in the form of boundary/interior layers and shock waves emerge as convection overwhelms diffusion. We show in this paper that these kinds of stiff problems can be well tackled by combining the LRBFCM with partial upwind scheme. For verification, several numerical examples are given to demonstrate that this scheme improves the LRBFCM in providing stable, accurate, and oscillation-free solutions to one- and two-dimensional Burgers׳ equations with shock waves and singular perturbation problems with turning points and boundary layers.

Based on the recently developed finite integration method (FIM) for solving one-dimensional partial differential equations by using the trapezoidal rule for numerical quadrature, we improve in this paper the FIM with an alternative extended Simpson׳s rule in which the Cotes and Lagrange formulas are used to determine the first order integral matrix. The improved one-dimensional FIM is then further extended to solve two-dimensional problems. Numerical comparison with the finite difference method and the FIM (Trapezoidal rule) are performed by several one- and two-dimensional real application including the Poisson type differential equations and plate bending problems. It has been shown that the newly revised FIM has made significant improvement in terms of accuracy compare without much sacrifice on the stability and efficiency.

The meshless local radial basis function method is applied to solve stationary and transient heat conduction problems in 2-D and 3-D bodies with functionally graded material properties. Time fractional derivative using Caputo definition is considered to describe anomalous diffusion phenomena. For temporal discretization, the Caputo time fractional derivative is approximated within each time interval 〈 t k , t k + 1 〉 by series of derivatives of integer order. The spatial discretization is performed by using the local radial basis collocation method. Numerical analyses are given on square (2D) and cubic (3D) domains to show the influence of the temporal fractional derivative parameter and gradation material parameter on the temperature distribution and temperature evolution in transient heat conduction problem.

Based on kernel-based approximation technique, we devise in this paper an efficient and accurate numerical scheme for solving a backward space-time fractional diffusion problem (BSTFDP). The kernels used in the approximation are the fundamental solutions of the space-time fractional diffusion equation expressed in terms of inverse Fourier transform of Mittag-Leffler functions. The use of Inverse fast Fourier transform (IFFT) technique enables an accurate and efficient evaluation of the fundamental solutions and gives a robust numerical algorithm for the solution of the BSTFDP. Since the BSTFDP is intrinsic ill-posed, we apply the standard Tikhonov regularization technique to obtain a stable solution to the highly ill-conditioned resultant system of linear equations. For choosing optimal regularization parameter, we combine the regularization technique with the generalized cross validation (GCV) method for an optimal placement of the source points in the use of fundamental solutions. Meanwhile, the proposed algorithm also speeds up the previous method given in Dou and Hon (2014). Several numerical examples are constructed to verify the accuracy and efficiency of the proposed method.

Based on the recently developed finite integration method for solving one-dimensional partial differential equation, we extend in this paper the method by using the technique of least squares to tackle higher-dimensional singular perturbation problems with multiple boundary layers. Theoretical convergence and numerical stability tests indicate that, even with the most simple numerical trapezoidal integration rule, the proposed method provides a stable, efficient, and highly accurate approximate solutions to the singular perturbation problems. An adaptive scheme on the refinement of integration points is also devised to better capture the stiff boundary layers. Illustrative examples are given in both 1D and 2D with comparison among some existing numerical methods.

Example-learning-based algorithms such as those based on sparse coding or neighbor embedding have been popular for single image super-resolution in recent years. However, affected by several critical factors on the training data and example representation, their reconstructions are usually plagued by kinds of artifacts. The removing of these artifacts is one of the major tasks for these methods. Unlike most existing methods that employ more complicated training methods, in this paper we would like to recover a clear reconstruction by fusing several "dirty" coarse reconstructions which are outputs of one or several simple training methods with small training set. One underlying key observation is that although coarse reconstructions are corrupted by different artifacts, they refer to the same high-resolution image. This global structure information is captured by an image structure-based low rank regularization method. The advantage of our method is that it can remove not only small noises but also gross artifacts. Except sparsity and randomness of the large artifacts, no other knowledge about them is required. Experimental results show that the proposed method can not only dramatically improve coarse reconstructions but also achieve competitive results.

We investigate in this paper an inverse problem (IP) of reconstructing inaccessible boundary values for parabolic equation with variable coefficients. The implicit finite difference (FD) for the IP is firstly introduced, which indicates different choices of the mesh-ratio compared with the same FD scheme for the direct problem. By means of the discrete Fourier transform, the FD scheme has a regularizing effect which prevents the solution from blowing up. Apart from the FD, a novel forward collocation (FC) method is formulated, which is based on the formulation of the IP into a sequence of well-posed direct problems and an ill-posed system of algebraic equations. The continuous dependence of a quasi-solution for the unknown boundary profile is firstly proven from which the existence of the quasi-solution in a compact set of admissible boundary profile is deduced. The corresponding dual problem is introduced with given proofs on the existence and error estimate of its quasi-solution. For numerical illustrat...

In this paper we investigate an inverse one-dimensional heat conduction problem in multi-layer medium. The inverse problem is first formulated in the frequency domain via Fourier transform technique. An effective regularization method for the stable reconstruction of solution is given with proven error estimates. Several numerical examples are constructed to demonstrate the effectiveness of the proposed method.

Based on the recently developed Finite Integration Method (FIM) for solving one-dimensional ordinary and partial differential equations, this paper extends the technique to higher dimensional partial differential equations. The main idea is to extend the first order finite integration matrices constructed by using either Ordinary Linear Approach (OLA) (uniform distribution of nodes) or Radial Basis Function (RBF) interpolation (uniform/random distributions of nodes) to higher order integration matrices. Using standard time integration techniques, such as Laplace transform, we have shown that the FIM is capable for solving time-dependent partial differential equations. Illustrative numerical examples are given in two-dimension to compare the FIM (FIM-OLA and FIM-RBF) with the finite difference method and point collocation method to demonstrate its superior accuracy and efficiency.

This paper explores the application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) for the solution of coupled heat transfer and fluid flow problems with a free surface. The method employs the representation of temperature, velocity and pressure fields on overlapping five-noded sub-domains through collocation by using Radial Basis Functions (RBFs). This simple representation is then used to compute the first and second derivatives of the fields from the respective derivatives of the RBFs. The energy and momentum equations are solved through explicit time integration scheme. For numerical efficiency, the Artificial Compressibility Method (ACM) with Characteristic Based Split (CBS) technique is firstly adopted to solve the pressure–velocity coupled equations. The performance of the method is assessed based on solving the classical two-dimensional De Vahl Davis steady natural convection benchmark problem with an upper free surface for Rayleigh number ranged from 103 to 105 and Prandtl number equals to 0.71.

We provide a class of positive definite kernels that allow to solve certain evolution equations of parabolic type for scattered initial data by kernel-based interpolation or approximation, avoiding time intergation completely. Some numerical illustrations are given.

Based on the recently developed local radial basis function method, we devise an implicit local radial basis function scheme, which is intrinsic mesh-free, for solving time fractional diffusion equations. In this paper the L1 scheme and the local radial basis function method are applied for temporal and spatial discretization, respectively, in which the time-marching iteration is performed implicitly. The robustness and accuracy of this proposed implicit local radial basis function method are demonstrated by the numerical example. Furthermore, the sensitivities of the shape parameter c and the number of nodes in the local sub-domain to the accuracy of numerical solutions are also investigated.

Using the heat equation as a simple example, we give a rigorous theoretical analysis of the Method of Lines, implemented as a meshless method based on spatial trial spaces spanned by translates of positive definite kernels. The technique can be generalized to other parabolic problems, and some numerical illustrations are given.

Based on the use of compactly supported radial basis functions, we extend in this paper the support vector approach to a multiscale support vector approach (MSVA) scheme for approximating the solution of a moderately ill-posed problem on bounded domain. The Vapnik's $$\epsilon $$∈-intensive function is adopted to replace the standard $$l^2$$l2 loss function in using the regularization technique to reduce the error induced by noisy data. Convergence proof for the case of noise-free data is then derived under an appropriate choice of the Vapnik's cut-off parameter and the regularization parameter. For noisy data case, we demonstrate that a corresponding choice for the Vapnik's cut-off parameter gives the same order of error estimate as both the a posteriori strategy based on discrepancy principle and the noise-free a priori strategy. Numerical examples are constructed to verify the efficiency of the proposed MSVA approach and the effectiveness of the parameter choices.

Based on kernel-based approximation technique, we devise in this paper an efficient and accurate numerical scheme for solving a backward problem of time-fractional diffusion equation (BTFDE). The kernels used in the approximation are the fundamental solutions of the time-fractional diffusion equation which can be expressed in terms of the M-Wright functions. To stably and accurately solve the resultant highly ill-conditioned system of equations, we successfully combine the standard Tikhonov regularization technique and the L-curve method to obtain an optimal choice of the regularization parameter and the location of source points. Several 1D and 2D numerical examples are constructed to demonstrate the superior accuracy and efficiency of the proposed method for solving both the classical backward heat conduction problem (BHCP) and the BTFDE.

A finite integration method is proposed in this paper to deal with partial differential equations in which the finite integration matrices of the first order are constructed by using both standard integral algorithm and radial basis functions interpolation respectively. These matrices of first order can directly be used to obtain finite integration matrices of higher order. Combining with the Laplace transform technique, the finite integration method is extended to solve time dependent partial differential equations. The accuracy of both the finite integration method and finite difference method are demonstrated with several examples. It has been observed that the finite integration method using either radial basis function or simple linear approximation gives a much higher degree of accuracy than the traditional finite difference method.

The finite integration method is proposed in this paper to approximate solutions of partial differential equations. The coefficient matrix of this finite integration method is derived and its superior accuracy and efficiency is demonstrated by making comparison with the classical finite difference method. For illustration, the finite integration method is applied to solve a nonlocal elastic straight bar under different loading conditions both for static and dynamic cases in which Laplace transform technique is adopted for the dynamic problems. Several illustrative examples indicate that high accurate numerical solutions are obtained with no extra computational efforts. The method is readily extendable to solve more complicated problems of nonlocal elasticity.

We investigate in this paper a non-characteristic Cauchy problem for a one-dimensional heat conduction equation in a multilayer domain, which is a well-known ill-posed problem. We first formulate the Cauchy problem in the frequency domain via Fourier transform technique. A modified version of the classical Tikhonov regularization technique together with the corresponding error estimates is presented. Several numerical examples are constructed to demonstrate the superior efficiency and accuracy of the proposed approach. Numerical results show that the modified Tikhonov regularization technique performs better than the classical Tikhonov regularization technique.

This paper investigates the solution of a parameter identification problem associated with the two-dimensional heat equation with variable diffusion coefficient. The singularity of the diffusion coefficient results in a nonlinear inverse problem which makes theoretical analysis rather difficult. Using an optimal control method, we formulate the problem as a minimization problem and prove the existence and uniqueness of the solution in weighted Sobolev spaces. The necessary conditions for the existence of the minimizer are also given. The results can be extended to more general parabolic equations with singular coefficients.

We investigate in this paper the stability of meshless unsymmetric collocation method by using radial basis functions for solving boundary value problems under Dirichlet, Neumann, or Robin boundary conditions. Using the monotonically decreasing property of the Fourier transforms of RBFs, we prove that the lowest bound of the resultant linear system depends on the separation distance of distinct centers and the decreasing order of the RBFs. Stability estimates can then be obtained for the meshless unsymmetric collocation method. For verification, several numerical examples are constructed to verify the theoretical results.

We investigate in this paper a Cauchy problem for the time-fractional diffusion equation (TFDE). Based on the idea of kernel-based approximation, we construct an efficient numerical scheme for obtaining the solution of a Cauchy problem of TFDE. The use of M -Wright functions as the kernel functions for the approximation space allows us to express the solution in terms of M -Wright functions, whose numerical evaluation can be accurately achieved by applying the inverse Laplace transform technique. To handle the ill-posedness of the resultant coefficient matrix due to the noisy Cauchy data, we adapt the standard Tikhonov regularization technique with the L -curve method for obtaining the optimal regularization parameter to give a stable numerical reconstruction of the solution. Numerical results indicate the efficiency and effectiveness of the proposed scheme.

In this paper we investigate an inverse problem for a time-fractional diffusion equation which is highly ill-posed in the two-dimensional setting. Based on an a priori assumption, we give a conditional stability result. Some new regularization methods are constructed for solving the inverse problem and the corresponding error estimates are proved. For numerical illustration, several examples are constructed to demonstrate the feasibility and efficiency of the proposed methods.

A meshless method based on the local Petrov–Galerkin approach is applied to inverse transient heat conduction problems in three-dimensional solids with continuously inhomogeneous and anisotropic material properties. The Heaviside step function is used as a test function in the local weak form, leading to the derivation of local integral equations. Nodal points are randomly distributed in the domain analyzed, and each node is surrounded by a spherical subdomain in which a local integral equation is applied. A meshless approximation based on the moving least-squares method is employed in the implementation. After performing spatial integrations, we obtain a system of ordinary differential equations for certain nodal unknowns. A backward finite-difference method is used for the approximation of the diffusive term in the heat conduction equation. A truncated singular-value decomposition is used to solve the ill-conditioned linear system of algebraic equations at each time step. The effectiveness of the meshless local Petrov–Galerkin (MLPG) method for this inverse problem is demonstrated by numerical examples.

We develop in this paper a new method to construct two explicit Lie algebras E and F. By using a loop algebra ? of the Lie algebra E and the reduced self-dual Yang?Mills equations, we obtain an expanding integrable model of the Giachetti?Johnson (GJ) hierarchy whose Hamiltonian structure can also be derived by using the trace identity. This provides a much simplier construction method in comparing with the tedious variational identity approach. Furthermore, the nonlinear integrable coupling of the GJ hierarchy is readily obtained by introducing the Lie algebra gN. As an application, we apply the loop algebra ? of the Lie algebra E to obtain a kind of expanding integrable model of the Kaup?Newell (KN) hierarchy which, consisting of two arbitrary parameters ? and ?, can be reduced to two nonlinear evolution equations. In addition, we use a loop algebra F? of the Lie algebra F to obtain an expanding integrable model of the BT hierarchy whose Hamiltonian structure is the same as using the trace identity. Finally, we deduce five integrable systems in R3 based on the self-dual Yang?Mills equations, which include Poisson structures, irregular lines, and the reduced equations.

This paper solves the Laplace equation Δu?=?0 on domains ?????3 by meshless collocation on scattered points of the boundary $\partial\Omega$ . Due to the use of new positive definite kernels K(x, y) which are harmonic in both arguments and have no singularities for x?=?y, one can directly interpolate on the boundary, and there is no artificial boundary needed as in the Method of Fundamental Solutions. In contrast to many other techniques, e.g. the Boundary Point Method or the Method of Fundamental Solutions, we provide a solid and comprehensive mathematical foundation which includes error bounds and works for general star-shaped domains. The convergence rates depend only on the smoothness of the domain and the boundary data. Some numerical examples are included.

In this paper the method of fundamental solutions (MFS) and the method of particular solution (MPS) are combined as a one-stage approach to solve the Cauchy problem for Poisson's equation. The main idea is to approximate the solution of Poisson's equation using a linear combination of fundamental solutions and radial basis functions. As a result, we provide a direct and effective meshless method for solving inverse problems with inhomogeneous terms. Numerical results in 2D and 3D show that our proposed method is effective for Cauchy problems.

A direct scheme is proposed to explicitly construct quasi-periodic wave solutions of the supersymmetric Ito's equation in superspace. This scheme is based on super Hirota bilinear formalism and a Riemann theta function formula. The relations between the periodic wave solutions and the well-known soliton solutions are rigorously established. Some new phenomena on super quasi-periodic waves are observed. For instance, there is an interesting influencing band occurred among the super quasi-periodic waves under the presence of the Grassmann variable. The super quasi-periodic waves are symmetric about the band but collapse along with the band.

A direct and unifying scheme for explicitly constructing quasiperiodic wave solutions (multiperiodic wave solutions) of supersymmetric KdV equation in a superspace is proposed. The scheme is based on the concept of super Hirota forms and on the use of super Riemann theta functions. In contrast to ordinary KdV equation with purely bosonic field, some new phenomena on super quasiperiodic waves occur in the supersymmetric KdV equation with the fermionic field. For instance, it is shown that the supersymmetric KdV equation does not possess an N-periodic wave solution for N≥ 2 for arbitrary parameters. It is further observed that there is an influencing band occurred among the quasiperiodic waves under the presence of the Grassmann variable. The quasiperiodic waves are symmetric about the band but collapse along with the band. In addition, the relations between the quasiperiodic wave solutions and soliton solutions are rigorously established. It is shown that quasiperiodic wave solution convergence to the soliton solutions under certain conditions and small amplitude limit.

Based on a Bäcklund transformation of the Riccati equation and its known soliton solutions, we obtain in this paper some exact traveling-wave solutions, including triangle function solutions and hyperbolic function solutions, of a hybrid lattice equation. The proposed method can be easily extended to locate exact solitary wave solutions for other types of differential-difference equations.

A systematic method for searching travelling-wave solutions to differential-difference equations (DDEs) is proposed in the paper. First of all, we introduce Backlund transformations for the standard Riccati equation which generate new exact solutions by using its simple and known solutions. Then we introduce a kind of formal polynomial solutions to DDEs and further determine the explicit forms by applying the balance principle. Finally, we work out exact solutions of the DDEs via substituting the form solutions and solving over-determined algebraic equations with the help of Maple. As illustrative examples, we obtain the travelling-wave solutions of the (2 + 1)-dimensional Toda lattice equation, the discrete modified KdV (mKdV) equation, respectively.

Based on the idea of radial basis functions approximation and the method of particular solutions, we develop in this paper a new meshless computational method to solve nonhomogeneous backward heat conduction problem. To illustrate the effectiveness and accuracy of the proposed method, we solve several benchmark problems in both two- and three-dimensions. Numerical results indicate that this novel approach can achieve an efficient and accurate solution even when the final temperature data is almost undetectable or disturbed with large noises. It has also been shown that the proposed method is stable to recover the unknown initial temperature from scattered final temperature data.

In this paper we employ a rational expansion to generalize Fan's method for exact travelling wave solutions for nonlinear partial differential equations (PDEs). To verify the reliability of the proposed method, the generalized shallow water wave (GSWW) equation has been investigated as an example. Kinds of new exact travelling wave solutions of a rational form have been obtained. This indicates that the proposed method provides a more general result for exact solution of nonlinear equations.

Based on the use of Green's function, we propose in this paper a new approach for solving specific classes of inverse source identification problems. Effective numerical algorithms are developed to recover both the intensities and locations of unknown point sources from scattered boundary measurements. For numerical verification, several boundary value problems defined on both bounded and unbounded regions of regular shape are given. Due to the use of closed analytic form of Green's function, the efficiency and accuracy of the proposed method can be guaranteed.

In this paper we consider complex coupled Higgs field equation and coupled Schrodinger–Boussinesq equation. An algebraic method is applied to construct solitary wave solutions, Jacobi periodic wave solutions and a range of other solutions of physical interest. It is shown that the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition.

The meshless local Petrov–Galerkin (MLPG) method is used to solve the inverse fracture problems in two-dimensional (2D) piezoelectric body. Electrical boundary conditions on the crack surfaces are not specified due to unknown dielectric permittivity of the medium inside the crack. Both stationary and transient dynamic boundary conditions are considered here. The analyzed domain is covered by small circular subdomains surrounding nodes spread randomly over the analyzed domain. A unit step function is chosen as test function in deriving the local integral equations (LIE) on the boundaries of the chosen subdomains. The Laplace-transform technique is applied to eliminate the time variation in the governing equation. The local integral equations are nonsingular and take a very simple form. The spatial variation of the Laplace transforms of displacements and electrical potential are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares (MLS) method. The singular value decomposition (SVD) is applied to solve the ill-conditioned linear system of algebraic equations obtained from the LIE after MLS approximation. The Stehfest algorithm is applied for the numerical Laplace inversion to retrieve the time-dependent solutions.

In this paper we first approximate a ‘nearly singular’ function, which tends to be the Dirac-delta function, to high degree of accuracy by using a recently developed Delta-shaped basis function. The Hermite-based meshless collocation method based on radial basis functions is then applied to solve a default barrier model, which is a time-dependent boundary value problem with a singularity at the initial condition. For numerical verification on the accuracy and efficiency of the newly proposed method, we compare the results with an analytical solution of the default barrier model under an assumption on the affine boundary. Numerical results indicate that the proposed method has potential advantage to solve problems with Dirac-type singularities.

With the help of two different Lie algebras and the corresponding loop algebras, the first and second kind of expanding integrable models of a new soliton hierarchy of evolution equations are obtained, respectively. The Hamiltonian structure of the first one is worked out by the quadratic-form identity. The bi-Hamiltonian structure of the second one is also generated. From the paper, we conclude that various Lie algebras really produce different soliton hierarchies of evolution equations. The approach presented in the paper provides a way for generating different integrable soliton expanding systems of the known soliton hierarchy of equations.

Based on the discrepancy principle, we develop in this paper a new method of choosing the location of source points to solve the backward heat conduction problem (BHCP) by using the method of fundamental solutions (MFS). The standard Tikhonov regularization technique with the L curve method for an optimal regularized parameter is adopted for solving the resultant highly ill-conditioned system of linear equations. Numerical verifications of the proposed computational method are presented for both the one-dimensional and the two-dimensional BHCP.

Based on a spectral problem and the Lenard operator pairs, we derive in this paper a modified Toda lattice hierarchy. The modified Toda lattice equation is first decomposed into systems of integrable ordinary differential equations. A hyper-elliptic Riemann surface and Abel–Jacobi coordinates are then introduced to linearize the associated flow, from which some quasi-periodic solutions of the modified Toda lattice can be explicitly constructed in terms of Riemann theta functions by using Jacobi inversion technique.

This paper investigates dynamic responses of a viscous fluid flow introduced under a time dependent pressure gradient in a rigid cylindrical tube that is lined with a deformable porous surface layer. With the Darcy’s law and a linear elasticity assumption, we have solved the coupling effect of the fluid movement and the deformation of the porous medium in the Laplace transform space. Governing equations are deduced for the solid displacement and the fluid velocity in the porous layer. Analytical solutions in the transformed domain are derived and the time dependent variables are inverted numerically using Durbin’s algorithm. Interaction between the solid and the fluid phases in the porous layer and its effects on fluid flow in tube are investigated under steady and unsteady flow conditions when the solid phase is either rigid or deformable. Examples are presented for flows driven by a Heaviside or a sinusoid pressure gradient. Significant effects of the porous surface layer on the flow in the tube are observed. The analytical solutions can be used to test more complicated numerical schemes.