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An efficient numerical method is proposed for the solution of the nonlinear cubic Schrodinger equation. The proposed method is based on the Frechet derivative and the meshless method with radial basis functions. An important characteristic of the method is that it can be extended from one-dimensional problems to multi-dimensional ones easily. By using the Frechet derivative and Newton–Raphson technique, the nonlinear equation is converted into a set of linear algebraic equations which are solved iteratively. Numerical examples reveal that the proposed method is efficient and reliable with respect to the accuracy and stability.

In the present work, local radial basis function (RBF) collocation method is used to solve the Stokes problem numerically. Benefits of the local meshless method is its flexibility of handling complex computational domain and interface. Local meshless method produces sparse coefficient matrix, though its structure is depending on the placement of nodes, and type of boundary conditions. The Stokes equations are collocated in a rectangular region with different interface conditions and direct solvers are used instead of iterative algorithm. Due to the unknown boundary conditions for the pressure term in the Stokes equations, value of the pressure is obtained by integrating the Stokes equations and eliminating the constant of integration. In the numerical implementation, circular, elliptic and Deltoid like interfaces are taken into consideration to test performance of the proposed approach for complex problems. Meshless solution of the Stokes equations for the first two problems are compared with the published results in the literature. The last problem is different form the first two problems in the sense that the continuity equation is also discontinuous and the velocity has nonzero value in both the inner and outer sub-domains.

This paper is concerned with the numerical solution of time- fractional partial differential equations (PDEs) via local meshless differential quadrature collocation method (LMM) using radial basis functions (RBFs). For the sake of comparison, global version of the meshless method is also considered. The meshless methods do not need mesh and approximate solution on scattered and uniform nodes in the domain. The local and global meshless procedures are used for spatial discretization. Caputo derivative is used in the temporal direction for both the values of \begin{document}$ \alpha \in (0,1) $\end{document} and \begin{document}$ \alpha\in(1,2) $\end{document} . To circumvent spurious oscillation casued by convection, an upwind technique is coupled with the LMM. Numerical analysis is given to asses accuracy of the proposed meshless method for one- and two-dimensional problems on rectangular and non-rectangular domains.

In this findings, a numerical meshless solution algorithm for 1D oscillatory Fredholm integral equation (OFIE) is put forward. The proposed algorithm is based on Levin?s quadrature theory (LQT) incorporating multi-quadric radial basis function (MQ-RBF). The procedure involves local approach of MQ-RBF differentiation matrix. The proposed method is specially designed to handle the case when the kernel function (KF) involves stationary point(s) (SP(s)). In addition to that, the model without SP(s) is also considered. The main advantage of the meshless procedure is that it can be easily extended to multidimensional geometry. These models have several physical applications in the area of engineering and sciences. The existence of the SP(s) in such models has numerous applications in the field of scattering and acoustics etc. (see [1, 2, 4, 6-8]). The proposed meshless method is accurate and cost-effective and provides a trustworthy platform to solve OFIE(s).

The main objective of this study is to numerically investigate the dynamical behavior of nonlinear Fitzhugh–Nagumo and Bateman–Burger systems through the Sinc collocation method by means of the $$\theta $$ -weighted scheme on various grid points of time-dependent evolutionary one spatial dimension in open quantum flow field model. The proposed technique based on the Sinc function is treated as a shape function to transform the governing nonlinear partial differential equation into an algebraic system. To approximate the time and spatial derivatives, finite difference scheme and the $$\theta $$ -weighted scheme have been used simultaneously due to the occurrence of infinite domain and multiple singularities. The effectiveness of the proposed results on the computational ground is illustrated graphically for better understanding and reliable performance of the design scheme is endorsed based on assessments of achieved accuracy in terms of stability analysis and convergence analysis.

In this research paper, an innovative bio-inspired algorithm based on evolutionary cubic splines method (CSM) has been utilized to estimate the numerical results of nonlinear ordinary differential equation Painlevé-I. The computational mechanism is used to support the proposed technique CSM and optimize the obtained results with global search technique genetic algorithms (GAs) hybridized with sequential quadratic programming (SQP) for quick refinement. Painlevé-I is solved by the proposed technique CSM-GASQP. In this process, variation of splines is implemented for various scenarios. The CSM-GASQP produces an interpolated function that is continuous upto its second derivative. Also, splines proved to be stable than a single polynomial fitted to all points, and reduce wiggles between the tabulated points. This method provides a reliable and excellent procedure for adaptation of unknown coefficients of splines by searching globally exploiting the performance of GA-SQP algorithms. The convergence, exactness and accuracy of the proposed scheme are examined through the statistical analysis for the several independent runs.

This paper focuses on the utilization of local radial basis functions (LRBFs) based level set method (LSM) for topology optimization of two-dimensional thermal problems using both concentrated as well as uniformly distributed heat generation. The design domain is embedded implicitly into a higher-dimensional function, which is parametrized with the LRBFs through an explicit scheme. This novel combination of LRBFs and LSM has the capability of controlling the topological variations automatically, i.e., hole insertion, merging with each other and with the boundary. The governing equations of heat conduction system are solved with the finite element method to obtain the sensitivities at level set grid points as a velocity field for evolution of the structural geometry. The objective function is set to the heat transfer potential with the maximum material volume as the design constraint. Several experiments are conducted on benchmark test problems and the resulting optimal solutions reveals efficiency, convergence and good agreement with those reported in the literature.

This paper studies Attilio Maccari (AM) system for some new solitary wave solutions. We used sub-ODE scheme to find kink shape, bell type, Weierstrass and Jacobi elliptic, hyperbolic, periodic and other solitary wave solutions under some constraint conditions. We will also present our results graphically in distinct dimensions.

A mixed-type formulation composed of Gauss-Laguerre quadrature and meshless collocation is presented for approximation of oscillatory integrals containing Hankel function of the first kind. An adaptive splitting procedure is implemented to resolve singularity of the transformed integral at x = 0. The singularity ridden integral is computed by the multi-resolution quadrature. Bessel function of the second kind is transformed into a Bessel function of the first kind before being approximated by the meshless collocation method (Zaman and Siraj-ul-Islam, J. Comput. Appl. Math. 315, 161–174, 2017). Theoretical error bounds of the proposed methods are established. Numerical results obtained from the benchmark problems are presented in the tabular and graphical forms for verification of theoretical error bounds and accuracy of the methods.

In the current paper, two numerical methods are proposed for the numerical solution of two- and three-dimensional elliptic partial differential equations (PDEs) with regular interfaces. The proposed methods are based on meshless collocation and Haar wavelet collocation. Numerical tests are performed to check accuracy and robustness of the proposed methods. Numerical results of the proposed methods are measured in terms of $$L_{\infty }$$ error norm to show their better accuracy than the existing methods.

Accurate algorithms are proposed for approximation of integrals involving highly oscillatory Bessel function of the first kind over finite and infinite domains. Accordingly, Bessel oscillatory integrals having high oscillatory behavior are transformed into oscillatory integrals with Fourier kernel by using complex line integration technique. The transformed integrals contain an inner non-oscillatory improper integral and an outer highly oscillatory integral. A modified meshfree collocation method with Levin approach is considered to evaluate the transformed oscillatory type integrals numerically. The inner improper complex integrals are evaluated by either Gauss–Laguerre or multi-resolution quadrature. Inherited singularity of the meshfree collocation method at x = 0 is treated by a splitting technique. Error estimates of the proposed algorithms are derived theoretically in the inverse powers of ω and verified numerically.

In the present paper, Multiquadric radial basis function (MQ RBF) and its integrated form are used to construct collocation methods for numerical solution of two-dimensional elliptic problems with curved or closed interface. The main purpose of this work is to perform a comparative analysis of both the methods via accuracy and condition number of the coefficient matrix for elliptic interface problems. In the classical RBF collocation method, the shape parameter is selected by using cross validation approach [1]. In the case of Integrated MQ RBF, a reasonable accuracy is obtained for a wide range of values of the shape parameter. Some of the benchmark problems such as linearized Poisson-Boltzmann problem [2], Poisson interface problem [3], Pennes Bioheat Equation [4] (with no exact solution, containing two phases), are considered to validate accuracy and efficiency of the RBFs collocation methods.

In this paper, two different Haar wavelet collocation multi-resolution procedures are proposed for linear partial differential equations (PDEs) with an unknown space-dependent heat source a...

In this paper, a Haar wavelet collocation method (HWCM) is developed for PDEs related to the framework of so-called inverse problem. These include PDEs with unknown time dependent heat source and unknown solution in interior of the domain. To this end, a transformation is used to eliminate the unknown heat source to obtain a PDE without a heat source. After elimination of unknown non-homogeneous term, an implicit finite-difference approximations is used to approximate the time derivative and Haar wavelets are used for approximation of the space derivatives. Several numerical experiments related to one- and two-dimensional heat sources are included to validate small condition number of coefficient matrix, accuracy and simple applicability of the proposed approach.

In this paper, a meshless solution procedure incorporating delaminating quadrature method for two-dimensional highly oscillatory Fredholm integral equation is put forward. The proposed method is an extension of our earlier findings of meshless methods for two-dimensional oscillatory Fredholm integral equation having kernel function free of stationary phase point(s) (Siraj-ul-Islam et al., 2015). The current method deals not only with the kernels having no stationary phase point(s) but also with the kernels having stationary phase point(s) in the context of highly oscillatory integral equations. The new method is numerically stable and computationally fast. Numerical experiments are presented to testify the robustness and efficiency of the proposed method.

In this paper, efficient and simple algorithms based on Levin’s quadrature theory and our earlier work involving local radial basis function (RBF) and Chebyshev differentiation matrices, are adopted for numerical solution of one-dimensional highly oscillatory Fredholm integral equations. This work is focused on the comparative performance of local RBF meshless and pseudospectral procedures. We have tested the proposed methods on phase functions with and without stationary phase point(s), both on uniform and Chebyshev grid points. The proposed procedures are shown accurate and efficient, and therefore provide a reliable platform for the numerical solution of integral equations. From the numerical results, we draw some conclusions about accuracy, efficiency and robustness of the proposed approaches.

Inverse heat problems are known to be ill-posed, which make them difficult to be solved numerically. In this paper we use meshless methods based on local and global radial basis functions for numerical solution of one- and two-dimensional inverse heat problems. In one-dimensional case, the unknown heat source and the right boundary condition are recovered, while in two-dimensional case, only the unknown heat source is recovered. Comparison of the global and the local meshless numerical schemes based on MQ radial basis function is performed for one- and two-dimensional cases. Several test problems related to inverse heat parabolic PDEs are numerically solved to verify accuracy and efficiency of the local meshless method. Without application of any regularization technique, accurate and stable solution has been obtained in the case of local meshless method for the input data, which is contaminated with a comparatively large level of noise.

In this paper, a local meshless differential quadrature collocation method is utilized to solve multi-dimensional reaction–convection–diffusion PDEs numerically. In some cases, global version of the meshless method is considered as well. The meshless methods approximate solution on scattered and uniform nodes in both local and global sense. In the case of convection-dominated PDEs, the local meshless method is coupled with an upwind technique to avoid spurious oscillations. For this purpose, a physically motivated local domain is utilized in the flow direction. Both regular and irregular geometries are taken into consideration. Numerical experiments are performed to demonstrate effective applications and accuracy of the meshless method on regular and irregular domains.

This paper introduces a weak meshless procedure combined with a multi-resolution numerical integration and its comparison with a strong local meshless formulation for approximating displacement and strain modeled in the form of Elliptic Boundary Value Problems (EBVPs) in one- and two-dimensional spaces. Assets and losses of both strong and weak meshless approaches are considered in detail. The meshless weak formulation considered in the current paper is the well-known Element Free Galerkin (EFG) method whereas the Local Radial Basis Functions Collocation Method (LRBFCM) is taken as a strong formulation. First aspect of the current work is implementation of the new numerical integration techniques introduced in Siraj-ul-Islam et al. (2010) and Aziz et al. (2011) [1,2] in the EFG method and its comparison with numerical integration based on standard Gaussian quadrature, adaptive integration and stabilized nodal integration techniques used in the context of EFG and other allied weak meshless formulations. Second aspect of the current work is analysis of comparative performance of the localized versions of strong and weak meshless formulations. Standard numerical tests are conducted to validate performance of both the approaches.

In this paper, new algorithms are proposed for numerical evaluation of multivariate highly oscillatory integrals having critical point(s) over piecewise smooth rectangular and non-rectangular domai...

In this paper, new quadrature rules are proposed for numerical evaluation of highly oscillatory integrals containing first kind of Bessel functions J ź ( ź x ) . Meshless procedure with uniform and scattered nodes is used to cope with frequent irregular oscillations caused by Bessel and Bessel-trigonometric functions. In addition, multi-resolution quadrature rules based on hybrid functions and Haar wavelets are used as supporting tools to handle the case of singularity of the meshless collocation method. Error bounds of the proposed methods are calculated and numerically verified by solving some benchmark test problems.

In this study, a computational intelligence technique based on three different designs of artificial neural networks (ANNs) is presented to solve the nonlinear Troesch’s boundary value problem arising in plasma physics. The structure of three ANN models is formulated by incorporating log-sigmoid (ANN-LS), radial-base (ANN-RB) and tan-sigmoid (ANN-TS) kernel functions in the hidden layers. Mathematical modeling of the problem is constructed for all three feed-forward ANN models by defining an error function in an unsupervised manner. Sequential quadratic programming method is employed for the learning of unknown parameters for all three ANN-LS, ANN-RB and ANN-TS schemes. The proposed models are applied to solve variants of Troesch’s problems by taking the small and large values of critical parameter in the system. A comparison of the proposed solution obtained from these models has been made with the standard numerical results of Adams method. The accuracy and convergence of the proposed models are investigated through results of statistical analysis in terms of performance indices based on the mean absolute deviation, root-mean-square error and variance account for.

In this article, the artificial intelligence techniques have been used for the solution of Falkner–Skan (FS) equations based on neural networks optimized with three methods including active set technique, sequential quadratic programming and genetic algorithms (GA) hybridization. Log-sigmoid activation function is used in artificial neural network architecture. The proposed techniques are applied to a number of cases for Falkner–Skan problems, and results were compared with GA hybrid results in all cases and were found accurate. The level of accuracy is examined through statistical analyses based on a sufficiently large number of independent runs.

A local meshless radial basis function collocation differential quadrature (LMRBFCDQ) is proposed for the numerical solution of a single and multi-asset option pricing PDE models arising in computational finance. Spatial discretization is performed by both local and a standard global meshless collocation procedures coupled with a set of different time integrators based on the forward Euler difference formula (FEDF), the fully Implicit method (FIM), the Crank–Nicolson method (CNM), the explicit Runge–Kutta method of order two (ERK2), the Crank–Nicolson Runge–Kutta method of order two (CNRK2), the fully Implicit Runge–Kutta method of order two (IRK2), the Runge–Kutta method of order four (RK4), the Embedded Runge–Kutta method (RK23). Operator splitting techniques like the ordinary operator splitting (OOS), the Lie–Trotter splitting, the additive splitting and the Strang splitting are also tested for time integration. The proposed hybrid schemes are the amalgamation of the meshless differential quadrature procedure and the finite difference approximations. Different types of radial basis functions (RBFs) i.e. the multiquadric (MQ), the inverse quadric (IQ) and the Gaussian (GA) are utilized for the spatial discretization of the PDE models. Numerical analysis of a range of computational finance related models are shown to demonstrate accuracy, efficiency and ease of implementation of the proposed meshless-finite difference procedure.

Most of the biological processes occurring in nature exhibit physical movements at macro and micro levels. These movements are convection–diffusion type and are studied due to the important role they play in the mathematical modeling and simulation of dynamical systems. Accurate simulation of complicated dynamical system models are quite challenging. The focus of the current work is to design an effective meshless procedure for the simulation of reaction–convection–diffusion type of epidemiological models. The simulation results of the proposed method show much realistic finite time blow-up, which occurs due to calamitous spatial movements of the susceptible class of the population. Numerical simulations obtained through the proposed method are carefully vetted and found consistent with the earlier results. In some cases, improvement in the earlier results is reported as well.

Based on Haar wavelet a new numerical method for the solution of a class of nth-order boundary-value problems (BVPs) with nonlocal boundary conditions is proposed. This new method is an extension of the Haar wavelet method (Siraj-ul-Islam et al. in Math Comput Model 50:1577---1590, 2010; Int J Therm Sci 50:686---697, 2011) from BVPs with local boundary conditions to BVPs with a class of nonlocal boundary conditions. A major advantage of the proposed method is that it is applicable to both linear and nonlinear BVPs. The method is tested on BVPs of third- and fourth-order. The numerical results are compared with an existing method in the literature and analytical solutions. The numerical experiments demonstrate the accuracy and efficiency of the proposed method.

In this work, we present a new method for solving elliptic partial differential equations using Haar wavelet. This work improves the earlier work (Aziz et al. in Appl Math Model 37:676–694, 2013) in terms of efficiency and contains an extension to nonlinear elliptic partial differential equations as well. In this paper the earlier algorithm (Aziz et al. in Appl Math Model 37:676–694, 2013) has been modified by starting the approximation with a fourth order mixed derivative rather than approximation of the second order derivatives with respect to x and y separately which results in a more efficient algorithm than the earlier algorithm. The use of Kronecker tensor products makes the new algorithm robust and easier to implement in a programming language. A distinguishing feature of the new method is that it can be applied to a variety of boundary conditions with a little modification of the program. The method is tested on several benchmark linear as well as nonlinear models. The numerical results show convergence, simple applicability and efficiency of the method.

A modified form of Levin’s method based on multi-quadric radial basis function is presented for numerical evaluation of integrals having squared oscillatory Bessel function of the first kind of order μ and oscillatory Airy function. The method converts the oscillatory integrals into a system of coupled ordinary differential equations (ODEs) and subsequently, numerical solution of the coupled ODEs is obtained by meshless collocation procedure. A multi-resolution quadrature is used to tackle singularity of the proposed method. Theoretical error analysis of the proposed methods is performed in asymptotic sense. Numerical test problems are included to verify accuracy and efficiency of the new methods in the context of oscillatory Airy and Bessel integrals.

In this paper, accurate and efficient algorithms based on global and local meshless formulations with modified Levin׳s quadrature are proposed for numerical solution of two-dimensional highly oscillatory Fredholm integral equations. The main focus of the study is to extend applications of global and local meshless procedures to integral equations on uniform as well as irregular nodes. This work will also focus on undertaking a comparative performance of the global and local meshless methods. The proposed methods will be tested on different test problems having phase function free of stationary points. From the numerical results we will draw some conclusions about accuracy and robustness of the proposed approaches. Further, the computational cost and the singularity of the system matrices in terms of condition number of both the local and global meshless methods will be investigated. A comprehensive analysis will ultimately establish computational efficiency and accuracy of the local approach.

In this paper a stable and modified form of the Levin method based on Bessel radial basis functions is employed for numerical solution of highly oscillatory integrals. In the proposed technique, the multiquadric radial basis function (Levin, 1982 [1] ; Siraj-ul-Islam et al., 2013 [2] ) is replaced by Bessel radial basis functions (Fornberg et al., 2006 [3] ) and thin plate spline of order three. In this scheme the integration form is first transformed into differential form and then the numerical solution of the corresponding differential form is found. The accuracy and the algebraic stability in the form of well-conditioned coefficient matrices of the proposed methods are confirmed through numerical experiments.

In the present paper, two numerical methods are analyzed for the solution of two-dimensional Poisson equation with two different types of nonlocal boundary conditions. The first numerical method is a collocation method based on Haar wavelet whereas the second numerical method is a meshless method based on different types of radial basis functions (RBFs). A two-point boundary condition and an integral boundary condition are the two types of nonlocal boundary conditions considered in the present work. For the collocation method based on Haar wavelet a new approach is formulated which involves the approximation of a fourth order mixed derivative by a Haar expansion which is integrated subsequently to get wavelet approximation of the solution. For the meshless method based on RBFs, the algorithm is implemented using two different splitting schemes (with and without shape parameter splitting) for numerical solution of the model. The comparative analysis of the meshless methods with and without shape parameter splitting scheme is performed between themselves as well as with the Haar wavelet. Accuracy and efficiency wise performance is confirmed through application of the algorithms on the benchmark tests.

A new numerical method based on Haar wavelet is proposed for two-dimensional nonlinear Fredholm, Volterra and Volterra-Fredholm integral equations of first and second kind. The proposed method is an extension of the Haar wavelet method Aziz and Siraj-ul-Islam (2013), Siraj-ul-Islam et al. (2013) and Siraj-ul-Islam et al. (2014) from one-dimensional nonlinear integral equations (Fredholm and Volterra) to two-dimensional nonlinear integral equations (Fredholm, Volterra and Volterra-Fredholm). The main characteristic of the method is that, unlike several other methods, it does not involve numerical integration which results in an improved accuracy of the method. In order to show the effectiveness of the method, it is applied to several benchmark problems. The numerical results are compared with other methods existing in the recent literature.

The PWR experimental benchmark problem defined by ANS was analyzed using WIMSD and PRIDE codes. Different modeling methodologies were used to calculate the infinite and effective multiplication factors. Relative pin power distributions were calculated for infinite lattice and critical core configurations, while reaction ratios were calculated for infinite lattice only. The discrete ordinate method (DSN) and collision probability method (PERSEUS) were used in each calculation. Different WIMSD cross-section libraries based on ENDF/B-VI.8, ENDF/B-VII.0, IAEA, JEF-2.2, JEFF-3.1 and JENDL-3.2 nuclear data files were also employed in the analyses. Comparison was made with experimental data and other reported results in order to find a suitable strategy for PWR analysis.

A new topology optimisation algorithm is implemented and presented for compliance minimisation of continuum structures using a volume preserving mechanism which effectively handles a volume constraint. The volume preserving mechanism is based on a unique combination of the level set method and a boundary element based bi-directional evolutionary structural optimisation approach using a bisectioning algorithm. The evolving structural geometry is implicitly represented with the level sets, efficiently handling complex topological shape changes, including holes merging with each other and with the boundary. Numerical results for two-dimensional linear elasticity problems suggest that the proposed adaptation provides smooth convergence of the objective function and a more robust, smoother geometrical description of the optimal design. Moreover, this new implementation allows for efficient material re-distribution within the design domain such that the objective function is minimised at constant volume. The proposed volume preserving mechanism can be easily extended to three-dimensional space.

In this paper, stochastic techniques have been developed to solve the 2-dimensional Bratu equations with the help of feed-forward artificial neural networks, optimized with particle swarm optimization (PSO) and sequential quadratic programming (SQP) algorithms. A hybrid of the above two algorithms, referred to as the PSO-SQP method is also studied. The original 2-dimensional equations are solved by first transforming them into equivalent one-dimensional boundary value problems (BVPs). These are then modeled using neural networks. The optimization problem for training the weights of the network has been addressed using particle swarm techniques for global search, integrated with an SQP method for rapid local convergence. The methodology is evaluated by applying on three different test cases of BVPs for the Bratu equations. Monte Carlo simulations and extensive analyses are carried out to validate the accuracy, convergence and effectiveness of the schemes. A comparative study of proposed results is made with available exact solution, as well as, reported numerical results.

In this paper, an operator splitting method based on meshless and finite difference procedures, is being considered for numerical solution of compartmental epidemiological population models with and without diffusion. A one step explicit meshless procedure is also applied for the numerical solution of the nonlinear model. The compartmental model contains susceptible, vaccinated, exposed, infected, and recovered (SVEIR) classes of the population. Effects of the diffusion on the simulation results of the model are being studied. Stability of endemic equilibrium point along with bifurcation analysis has also been investigated. Due to non-availability of the exact solution, the numerical results obtained are mutually compared and their correctness is being verified by the theoretical results as well.

In this paper, a novel technique is being formulated for the numerical solution of integral equations (IEs) as well as integro-differential equations (IDEs) of first and higher orders. The present approach is an improved form of the Haar wavelet methods (Aziz and Siraj-ul-Islam, 2013, Siraj-ul-Islam et al., 2013). The proposed modifications resulted in computational efficiency and simple applicability of the earlier methods (Aziz and Siraj-ul-Islam, 2013, Siraj-ul-Islam et al., 2013). In addition to this, the new approach is being extended from IDEs of first order to IDEs of higher orders with initial- and boundary-conditions. Unlike the methods (Aziz and Siraj-ul-Islam, 2013, Siraj-ul-Islam et al., 2013) (where the kernel function is being approximated by two-dimensional Haar wavelet), the kernel function in the present case is being approximated by one-dimensional Haar wavelet. The modified approach is easily extendable to higher order IDEs. Numerical examples are being included to show the accuracy and efficiency of the new method.

In this paper we present two new numerically stable methods based on Haar and Legendre wavelets for one- and two-dimensional parabolic partial differential equations (PPDEs). This work is the extension of the earlier work [1–3] from one- and two-dimensional boundary-value problems to one- and two- dimensional PPDEs. Two generic numerical algorithms are derived in two phases. In the first stage a numerical algorithm is derived by using Haar wavelets and then in the second stage Haar wavelets are replaced by Legendre wavelets in quest for better accuracy. In the proposed methods the time derivative is approximated by first order forward difference operator and space derivatives are approximated using Haar (Legendre) wavelets. Improved accuracy is obtained in the form of wavelets decomposition. The solution in this process is first obtained on a coarse grid and then refined towards higher accuracy in the high resolution space. Accuracy wise performance of the Legendre wavelets collocation method (LWCM) is better than the Haar wavelets collocation method (HWCM) for problems having smooth initial data or having no shock phenomena in the solution space. If sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, LWCM loses its accuracy in such cases, whereas HWCM produces a stable solution in such cases as well. Contrary to the existing methods, the accuracy of both HWCM and LWCM do not degrade in case of Neumann’s boundary conditions. A distinctive feature of the proposed methods is its simple applicability for a variety of boundary conditions. Performances of both HWCM and LWCM are compared with the most recent methods reported in the literature. Numerical tests affirm better accuracy of the proposed methods for a range of benchmark problems.

A new method is proposed for numerical solution of Fredholm and Volterra integro-differential equations of second kind. The proposed method is based on Haar wavelets approximation. Special characteristics of Haar wavelets approximation has been used in the derivation of this method. The new method is the extension of the recent work [Aziz and Siraj-ul-Islam, New algorithms for numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comput. Appl. Math. 239 2013, pp. 333–345] from integral equations to integro-differential equations. The method is specifically derived for nonlinear problems. Two new algorithms are also proposed based on this new method, one each for numerical solution of Fredholm and Volterra integro-differential equations. The proposed algorithms are generic and are applicable to all types of both nonlinear Fredholm and Volterra integro-differential equations of second kind. The cost of the new algorithms is considerably reduced by using the Broyden's method instead of Newton's method for solution of system of nonlinear equations. Most of the numerical methods designed for solution of integro-differential equations rely on some other technique for numerical integration. The advantage of our method is that it does not use numerical integration. The integrand is approximated using Haar wavelets approximation and then exact integration is performed. The method is tested on number of problems and numerical results are compared with existing methods in the literature. The numerical results indicate that accuracy of the obtained solutions is reasonably high even when the number of collocation points is small.

In this paper, we formulate and employ efficient and accurate methods for numerical evaluation of highly oscillatory integrals and integrals having stationary points. Two new approaches using radial basis function (RBF) and wavelets are discussed. The first approach is related to meshless method (MM) which is based on multiquadric (MQ) RBF, and is specially designed for integrands having oscillatory character. This approach stems from the Levin's method. In this procedure, the solution is obtained by solving the corresponding ODE or PDE instead of finding a numerical solution of the integration problem. In situations when the integrand has stationary points, MM fails to deliver. We opt for quadrature rules based on Haar wavelets and hybrid functions. The proposed methods are tested on a number of benchmark tests considered in available literature. The performance of the new methods is compared with the existing methods. Better accuracy of the proposed methods is reported for a variety of problems.

This paper tackles an improved Localized Radial Basis Functions Collocation Method (LRBFCM) for the numerical solution of hyperbolic partial differential equations (PDEs). The LRBFCM is based on multiquadric (MQ) Radial Basis Functions (RBFs) and belongs to a class of truly meshless methods which do not need any underlying mesh. This method can be implemented on a set of uniform or random nodes, without any a priori knowledge of node to node connectivity. We have chosen uniform nodal arrangement due their suitability and better accuracy. Five nodded domains of influence are used in the local support for the calculation of the spatial partial derivatives. This approach results in a small interpolation matrix for each data center and hence the time integration has comparatively low computational cost than the related global method. Different sizes of domain of influence i.e. m=5,13 are considered. Shape parameter sensitivity of MQ is handled through scaling technique. The time derivative is approximated by first order forward difference formula. An adaptive upwind technique is used for stabilization of the method. Capabilities of the LRBFCM are tested by applying it to one- and two-dimensional benchmark problems with discontinuities, shock pattern and periodic initial conditions. Performance of the LRBFCM is compared with analytical solution, other numerical methods and the results reported earlier in the literature. We have also made comparison with implicit first order time discretization and first order upwind spatial discretization (FVM1) and implicit second order time discretization and first order upwind spatial discretization (FVM2) as well. Accuracy of the method is assessed as a function of time and space. Numerical convergence is also shown for both one- and two-dimensional test problems. It has been observed that the proposed method is more efficient in terms of less memory requirement and less computational efforts due to one time inversion of 5x5 (size of local domain of influence) coefficient matrix. The results obtained through LBRFCM are stable and comparable with the existing methods for a variety of problems with practical applications.

Based on collocation with Haar and Legendre wavelets, two efficient and new numerical methods are being proposed for the numerical solution of elliptic partial differential equations having oscillatory and non-oscillatory behavior. The present methods are developed in two stages. In the initial stage, they are developed for Haar wavelets. In order to obtain higher accuracy, Haar wavelets are replaced by Legendre wavelets at the second stage. A comparative analysis of the performance of Haar wavelets collocation method and Legendre wavelets collocation method is carried out. In addition to this, comparative studies of performance of Legendre wavelets collocation method and quadratic spline collocation method, and meshless methods and Sinc–Galerkin method are also done. The analysis indicates that there is a higher accuracy obtained by Legendre wavelets decomposition, which is in the form of a multi-resolution analysis of the function. The solution is first found on the coarse grid points, and then it is refined by obtaining higher accuracy with help of increasing the level of wavelets. The accurate implementation of the classical numerical methods on Neumann’s boundary conditions has been found to involve some difficulty. It has been shown here that the present methods can be easily implemented on Neumann’s boundary conditions and the results obtained are accurate; the present methods, thus, have a clear advantage over the classical numerical methods. A distinct feature of the proposed methods is their simple applicability for a variety of boundary conditions. Numerical order of convergence of the proposed methods is calculated. The results of numerical tests show better accuracy of the proposed method based on Legendre wavelets for a variety of benchmark problems.

Two new algorithms based on Haar wavelets are proposed. The first algorithm is proposed for the numerical solution of nonlinear Fredholm integral equations of the second kind, and the second for the numerical solution of nonlinear Volterra integral equations of the second kind. These methods are designed to exploit the special characteristics of Haar wavelets in both one and two dimensions. Formulae for calculating Haar coefficients without solving the system of equations have been derived. These formulae are then used in the proposed numerical methods. In contrast to other numerical methods, the advantage of our method is that it does not involve any intermediate numerical technique for evaluation of the integral present in integral equations. The methods are validated on test problems, and numerical results are compared with those from existing methods in the literature.

In this paper, numerical treatment is presented for the solution of boundary value problems of one-dimensional Bratu-type equations using artificial neural networks. Three types of transfer functions including Log-sigmoid, radial basis, and tan-sigmoid are used in the neural networks’ modeling. The optimum weights for all the three networks are searched with the interior point method. Various test cases of Bratu-type equations have been simulated using the developed models. The accuracy, convergence, and effectiveness of the methods are substantiated by a large number of simulation data for each model by taking enough independent runs.

A comparison of the performance of the global and the local radial basis function collocation meshless methods for three dimensional parabolic partial differential equations is performed in the present paper. The methods are structured with multiquadrics radial basis functions. The time-stepping is performed in a fully explicit, fully implicit and Crank–Nicolson ways. Uniform and non-uniform node arrangements have been used. A three-dimensional diffusion–reaction equation is used for testing with the Dirichlet and mixed Dirichlet–Neumann boundary conditions. The global methods result in discretization matrices with the number of unknowns equal to the number of the nodes. The local methods are in the present paper based on seven-noded influence domains, and reduce to discretization matrices with seven unknowns for each node in case of the explicit methods or a sparse matrix with the dimension of the number of the nodes and seven non-zero row entries in case of the implicit method. The performance of the methods is assessed in terms of accuracy and efficiency. The outcome of the comparison is as follows. The local methods show superior efficiency and accuracy, especially for the problems with Dirichlet boundary conditions. Global methods are efficient and accurate only in cases with small amount of nodes. For large amount of nodes, they become inefficient and run into ill-conditioning problems. Local explicit method is very accurate, however, sensitive to the node position distribution, and becomes sensitive to the shape parameter of the radial basis functions when the mixed boundary conditions are used. Performance of the local implicit method is comparatively better than the others when a larger number of nodes and mixed boundary conditions are used. The paper represents an extension of our recently made similar study in two dimensions.

In this paper, a stochastic technique is developed to solve 2-dimensional Bratu equations using feed-forward artificial neural networks, optimized with genetic and interior-point algorithms. The 2-dimensional equations are first transformed into a 1-dimensional boundary value problem, and a mathematical model of the transformed equation is then formulated with neural networks using an unsupervised error. Network weights are optimized to minimize the error. Evolutionary computing based on genetic algorithms is used as a tool for global search, integrated with an interior-point method for rapid local convergence. The methodology is applied to solve three cases of boundary value problems for the Bratu equations. The accuracy, convergence and effectiveness of the scheme is validated for a large number of simulations. Comparison of results is made with the exact solution derived using MATHEMATICA, and is found to be in good agreement.