# Amit Kumar VermaIndian Institute of Technology Patna | IIT Patna · Department of Mathematics

Amit Kumar Verma

PhD

## About

107

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681

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Citations since 2017

Introduction

In the era of the internet, social media where within few days news become viral, be careful. Anything which is built on a shallow foundation just to achieve the attention of people does not grow and stay long. Anything which has left a great impact on people or potential to leave a legacy behind takes lifetime effort. So stay focused.

Additional affiliations

December 2015 - February 2022

July 2009 - December 2015

Education

March 2003 - February 2009

## Publications

Publications (107)

In this paper, we develop numerical methods based on the fuzzy transform methods
(FTMs). In this approach we apply fuzzy transforms on discrete version of the
derivatives and use it to derive FTMs. We also establish convergence of the proposed
FTMs. To test the efficiency of the proposed FTMs, we apply the FTM schemes on the
second order nonlinear...

We consider the generalized Burgers Huxley equation (GBH) subject to certain initial and boundary conditions (BCs). Using a solitary wave solution, we derive an exact finite difference (EFD) scheme for the GBH equation. Furthermore, we propose a non-standard finite difference (NSFD) scheme which operates for all $\theta \in \mathbb{N}.$ The qualita...

In this article we consider the fourth order non-self-adjoint singular boundary value problem $$ \frac{1}{r}\left[ r \left\lbrace \frac{1}{r} \left(r \phi' \right)^{'} \right\rbrace^{'}\right]^{'}=\frac{\phi' \phi''}{r}+\lambda,$$ with $\lambda$ as a parameter measures the speed at which new particles are deposited.
This differential equation is...

Purpose
In this article, the authors consider the following nonlinear singular boundary value problem (SBVP) known as Lane–Emden equations, − u ″( t )-( α / t ) u ′( t ) = g ( t , u ), 0 < t < 1 where α ≥ 1 subject to two-point and three-point boundary conditions. The authors propose to develop a novel method to solve the class of Lane–Emden equati...

Solving Burgers' equation always posses challenges for a small value of viscosity. Here we present a numerical method based on the Haar wavelet collocation method coupled with a nonstandard finite difference (NSFD) scheme for a class of generalized Burgers' equation. In the solution process, the time derivative is discretized by the NSFD scheme and...

In this article, we give a new definition of the linear canonical Stockwell transform (LCST) and study its basic properties along with the inner product relation, reconstruction formula and also characterize the range of the transform and show that its range is the reproducing kernel Hilbert space. We also develop a multiresolution analysis (MRA) a...

In this work, we propose the following fourth‐order non‐self‐adjoint SBVPs to investigate 1rαrα1rα(rαϕ′)′′′=12rαμ′ϕ′2+2μϕ′ϕ″+λG(r),for0<r<1,$$ \frac{1}{r^{\alpha }}{\left\{{r}^{\alpha }{\left[\frac{1}{r^{\alpha }}{\left({r}^{\alpha }{\phi}^{\prime}\right)}^{\prime}\right]}^{\prime}\right\}}^{\prime }=\frac{1}{2{r}^{\alpha }}\left({\mu}^{\prime }{\p...

In this paper, we analyse the accuracy of the Haar wavelet approximation method on the singular boundary value problem. We propose to solve some higher-order nonlinear singular BVPs. We also verify that numerically the estimated order of convergence is in agreement with the obtained theoretical results.

In this article, we develop a computational technique for solving the nonlinear time-fractional one and two-dimensional partial integro-differential equation with a weakly singular kernel. For the approximation of spatial derivatives, we apply the Haar wavelets collocation method whereas, for the time-fractional derivative, we use the nonstandard f...

In this article, we analyze and propose to compute the numerical solutions of a generalized Rosenau–KDV–RLW (Rosenau‐Korteweg De Vries‐Regularlized Long Wa) equation based on the Haar wavelet (HW) collocation approach coupled with nonstandard finite difference (NSFD) scheme and quasilinearization. In the process of the numerical solution, the NSFD...

Capturing solution near the singular point of any nonlinear SBVPs is challenging because coefficients involved in the differential equation blow up near singularities. In this article, we aim to construct a general method based on orthogonal polynomials as wavelets, i.e., orthogonal polynomial wavelet method (OPWM). We also discuss the convergence...

This study considers three-point Hermite interpolation as an approximation method that utilizes the values of a function and its derivatives at the two endpoints and another point of the domain to reconstruct the function. Therefore, it can also be considered as a three-point Taylor expansion. In the process of conducting the research, a novel and...

This paper deals with the two classes of non-linear advection–diffusion–reaction (ADR) equations subject to certain initial and boundary conditions. We derive exact finite difference (FD) schemes with the help of solitary wave solutions of ADR equations. Furthermore, we construct non-standard FD schemes for both the equations. The positivity and bo...

In this article, we explore the monotone iterative technique (MI‐technique) to study the existence of solutions for a class of nonlinear Neumann four‐point, boundary value problems (BVPs) defined as, −ϕ(2)(z)=G(z,ϕ,ϕ(1)),0<z<1,ϕ(1)(0)=λϕ(1)(β1),ϕ(1)(1)=δϕ(1)(β2),$$ {\displaystyle \begin{array}{c}\hfill -{\phi}^{(2)}(z)=G\left(z,\phi, {\phi}^{(1)}\r...

In this paper, we extend the quadratic phase Fourier transform of a complex valued functions to that of the quaternion valued functions of two variables. We call it the quaternion quadratic phase Fourier transform (QQPFT). Based on the relation between the QQPFT and the quaternion Fourier transform (QFT) we obtain the sharp Hausdorff-Young inequali...

We propose two novel non-standard finite difference (NSFD) schemes for a class of non-linear singular boundary value problems (SBVPs). One of the basic idea of NSFD schemes is that the step size $\Delta t$ is replaced by a non-linear function, e.g., $\sin \Delta t$, $\tan\Delta t$ etc. In the present work, we propose to include one more parameter i...

In this errata sheet, we comment on the definition of the kernel of the continuous fractional wavelet transform (CFrWT) studied in the article “A certain family of fractional wavelet transformations” by Srivastava, Khatterwani and Upadhyay [Mathematical Methods in the Applied Sciences, Vol 42, No. 9, 3103–3122, 2019]. We have modified the definitio...

In this work, we focus on the following non-linear fourth order SBVP \begin{eqnarray} \nonumber \frac{1}{r}\left[ r \left\lbrace \frac{1}{r} \left(r \phi’ \right)^{’} \right\rbrace^{’}\right]^{’}=\frac{\phi’ \phi’‘}{r}+\lambda, \end{eqnarray} where $\lambda$ is a parameter. We convert this non-linear differential equation into third order non-linea...

We define a novel time-frequency analyzing tool, namely linear canonical wavelet transform (LCWT) and study some of its important properties like inner product relation, reconstruction formula and also characterize its range. We obtain Donoho-Stark's and Lieb's uncertainty principle for the LCWT and give a lower bound for the measure of its essenti...

In this paper, we have given a new definition of continuous fractional wavelet transform in $\mathbb{R}^N$, namely the multidimensional fractional wavelet transform (MFrWT) and studied some of the basic properties along with the inner product relation and the reconstruction formula. We have also shown that the range of the proposed transform is a r...

In this article, we develop a monotone iterative technique (MI-technique) with lower and upper (L-U) solutions for a class of four-point Dirichlet nonlinear boundary value problems (NLBVPs), defined as, ... where ..., ... the non linear term ...is continuous function in x, one sided Lipschitz in ψ and Lipschitz in . To show the existence result, we...

In the present paper, we establish an efficient numerical scheme based on weakly L-stable time integration convergent formula and nonstandard finite difference (NSFD) scheme. We solve Burger’s equation with Dirichlet boundary conditions as well as Neumann boundary conditions. We also solve the Fisher equation. We use Hermite approximation polynomia...

In this paper, an effective iterative technique for analytical solutions of a class of nonlinear singular boundary value problems (SBVPs) occurring in different physical situations is presented. In constructing the recursive approach for the iterative solution components, a technique that relies on establishing a corresponding integral representati...

This work examines the existence of the solutions of a class of three-point nonlinear boundary value problems that arise in bridge design due to its nonlinear behavior. A maximum and anti-maximum principles are derived with the support of Green’s function and their constant sign. A different monotone iterative technique is developed with the use of...

In this article, we explore the monotone iterative technique (MI-technique) to study the existence of solutions for a class of nonlinear Neumann 4-point, boundary value problems (BVPs) defined as, \begin{eqnarray*} \begin{split} -\z^{(2)}(\y)=\x(\y,\z,\z^{(1)}),\quad 0<\y<1,\\ \z^{(1)}(0)=\lambda \z^{(1)}(\beta_1 ),\quad \z^{(1)}(1)=\delta \z^{(1)}...

In this work, we focus on the non-linear fourth order class of singular boundary value problem which contains the parameter $\lambda$. We convert this non-linear differential equation into third order non-linear differential equation. The third order problem is singular, non self adjoint and nonlinear. Moreover, depending upon $\lambda$, it admits...

In this paper, we study continuous fractional wavelet transform (CFrWT) in $n$-dimensional Euclidean space $\mathbb{R}^n$ with scaling parameter $\boldsymbol a=(a_{1},a_{2},\ldots,a_{n}) \in \mathbb{R}^n$ such that $a_i\neq0,~\forall i=1(1)n$. We obtain inner product relation and reconstruction formula for the CFrWT depending on two wavelets along...

In this paper, we define a new class of continuous fractional wavelet transform (CFrWT) and study its properties in Hardy space and Morrey space. The theory developed generalize and complements some of the already existing results.

This paper deals with the nonlinear generalized advection-diffusion-reaction (GADR) equation subject to some initial and boundary conditions (BCs). Some exact finite difference (EFD) schemes and non-standard finite difference (NSFD) schemes are derived. Positivity and boundedness of the proposed NSFD schemes are analysed analytically and numericall...

This article aims to prove the existence of a solution and compute the region of existence for a class of four-point nonlinear boundary value problems (NLBVPs) defined as,
-u''(x)=\psi(x,u,u'), \quad x\in (0,1),
u'(0)=\lambda_{1}u(\xi), u'(1)=\lambda_{2} u(\eta),
where I=[0,1], 0<\xi\leq\eta<1 and \lambda_1,\lambda_2> 0. The non linear source term...

We propose a new class of SBVPs which deals with exothermic reactions. We
also propose four computationally stable methods to solve singular nonlinear
BVPs by using Hermite wavelet collocation which are coupled with Newton’s
quasilinearization and Newton-Raphson method. We compare the results
which are obtained by using Hermite wavelets with the re...

In this work, the existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. Since we are interested in radial solutions, we focus on the fourth-order singular ordinary differential equation. It is non-self adjoint, it does not have exact solutions, and...

In this paper we define a new class of continuous fractional wavelet transform (CFrWT) and study its properties in Hardy space and Morrey space. The theory developed generalize and complement some of already existing results.

In this article, we come up with a novel numerical scheme based on Haar wavelet (HW) along with nonstandard finite difference (NSFD) scheme to solve time-fractional Burgers’ equation with variable diffusion coefficient and time delay. In the solution process, we discretize the fractional time derivative by NSFD formula and spatial derivative b...

In this paper we consider the following class of four point boundary value problems −y ' ' (x)=f (x , y) , 0< x< 1, y ' (0)=0, y (1)=δ 1 y (η 1) + δ 2 y (η 2) , where δ 1 , δ 2 ≥ 0 , 0<η 1 , η 2 <1, and f (x , y) is continuous in x one sided Lipschitz in y. Existence of solution is discussed in both theoretical and numerical manner. A monotone iter...

Solving Burgers' equation always posses challenges for small values of viscosity. Here we propose a method to compute the numerical solution for a class of generalised Burgers' equation based on the Haar wavelet (HW) coupled with nonstandard finite difference (NSFD) method. In the solution process, the time derivative is discretised by nonstandard...

As a generalization of Fourier transform (FT) and fractional Fourier transform (FrFT), linear canonical transform (LCT) is a three parameter family of integral transform. Compared to FT and FrFT it has more degree of freedom. But due to the presence of global kernel it is not capable of indicating the time localization of the LCT spectral component...

We consider a generalized form of Burgers Fisher (BF) equation subject to certain initial and boundary conditions. We propose an explicit exact finite difference (EFD) scheme for the BF equation using its solitary wave solution. Furthermore, a non-standard finite difference (NSFD) scheme is also proposed. The properties like positivity and boundedn...

In this paper, we study continuous fractional wavelet transform (CFrWT) in $\mathbb{R}^n$ with scaling parameter $\boldsymbol a=(a_{1},a_{2},\ldots,a_{n}) \in \mathbb{R}^n$. We derive some of its basic properties like inner product relation and inversion formula. We also characterize the range of the transform. Moreover, we also
study the boundedn...

Lane-Emden type equations arise various physical phenomena in mathematical and astrophysics like stellar structure, thermionic currents, thermal explosions, radiative cooling, CTC, etc. In this work, we consider a model by considering the equation
(x^{\beta}y'(x))'+x^{\beta}f(x,y)=0, 0<x<1,
y'(0)=0, b_{1}y(1)+a_{1}y'(1)=c_{1}.
For $\beta=1$ and $\b...

In this article, we propose novel coupled nonlinear singular boundary value problems arising in epitaxial growth theory. The coupled equations are nonlinear non-self-adjoint and singular and have no exact solutions. We derive some qualitative properties of the coupled solutions, which depend on the size of parameters that occur in the coupled syste...

Purpose - In this work, we propose an efficient computational technique which uses Haar wavelets collocation approach coupled with the Newton-Raphson method and solves the following class of system of Lane-Emden equations -(t^{k_1} y'(t))'=t^{-\omega_1} f_1(t,y(t),z(t)), -(t^{k_2} z'(t))'=t^{-\omega_2} f_2(t,y(t),z(t)), where $t>0$, subject to the...

In this paper, we propose a $7^{th}$ order weakly $L$-stable time integration scheme. In the process of derivation of the scheme, we use explicit backward Taylor's polynomial approximation of sixth-order and Hermite interpolation polynomial approximation of fifth order. We apply this formula in the vector form to solve Burger's equation which is a...

If there is a jump discontinuity present in the forcing term of a boundary value problem (BVP), the nonstandard finite difference (NSFD) and finite difference (FD) methods do not approximate the solutions very well. Here we use fuzzy transforms (FTs) and derive fuzzy transformed NSFD schemes that are referred to as non-standard fuzzy transform meth...

The paper concerns numerical study of a Stokes-Brinkman system with varying liquid viscosity that describes the fluid flow along a set of partially porous parallel cylindrical particles, which form a fibrous
membrane, using the cell modeling. We have applied two different approaches to varying viscosity inside a porous layer: exponential and power...

Several real-life problems are modeled by nonlinear singular differential equations. In this article, we study a class of nonlinear singular differential equations, explore its various aspects, and provide a detailed literature survey. Nonlinear singular differential equations are not easy to solve and their exact solution does not exist in most ca...

Purpose - In this work, we propose an efficient computational technique which uses Haar wavelets collocation approach coupled with the Newton-Raphson method and solves the following class of system of Lane-Emden equations
-(t^{k_1} y'(t))'=t^{-\omega_1} f_1(t,y(t),z(t)),
-(t^{k_2} z'(t))'=t^{-\omega_2} f_2(t,y(t),z(t)),
where $t>0$, subject to the...

A uniform Haar wavelet collocation based method is proposed for finding the numerical results for a class of third order nonlinear (Emden-Fowler type) singular differential equations with initial and boundary conditions. At the point of singularity, the coefficient of the such equation blows up, that causes difficulties in capturing the numerical s...

The existence of numerical solutions to a fourth order singular boundary value problem arising in the theory of epitaxial growth is studied. An iterative numerical method is applied on a second order nonlinear singular boundary value problem which is the exact result of the reduction of this fourth order singular boundary value problem. It turns ou...

Solving Burgers' equation always poses challenge to researchers as for small values of viscosity the analytical solution breaks down. Here we propose to compute numerical solution for a class of generalised Burgers' equation described as $$ \frac{\partial w}{\partial t}+ w^{\mu}\frac{\partial w}{\partial x_{*}}=\nu w^{\delta}\frac{\partial^2 w}{\pa...

In this paper, we have studied continuous fractional wavelet transform (CFrWT) in $n$-dimensional Euclidean space $\mathbb{R}^n$ with dilation parameter $\boldsymbol a=(a_{1},a_{2},\ldots,a_{n}),$ such that none of $a_{i}'s$ are zero. Necessary and sufficient condition for the admissibility of a function is established with the help of fractional c...

The aim of this article is to prove the existence of solution and compute the region of existence for a class of 4-point BVPs defined as, \begin{eqnarray*} &&-u''(x)=\psi(x,u,u'), \quad 0<x<1, &&u'(0)=\lambda_{1}u(\xi), \quad u'(1)=\lambda_{2} u(\eta), \end{eqnarray*} where $I=[0,1]$, $0<\xi\leq\eta<1$ and $\lambda_1,\lambda_2> 0$. The non linear s...

The regions of existence are established for a class of two point nonlinear diffusion type boundary value problems (NDBVP)
\begin{eqnarray*}
&&\label{abst-intr-1} -s''(x)-ns'(x)-\frac{m}{x}s'(x)=f(x,s), \qquad m>0,~n\in \mathbb{R},\qquad x\in(0,1),\\
&&\label{abst-intr-2} s'(0)=0, \qquad a_{1}s(1)+a_{2}s'(1)=C,
\end{eqnarray*}
where $a_{1}>0,$ $a_{...

In this note we establish existence of solutions of singular boundary value problem −(p(x)y ′ (x)) ′ = q(x)f (x, y, py ′) for 0 < x ≤ b and y ′ (0) = 0, α 1 y(b)+β 1 p(b)y ′ (b) = γ 1 with p(0) = 0 and q(x) is integrable. Regions of multiple solutions have also been determined.

In this paper we have considered generalized Emden-Fowler equation,
y''(t)+\sigma t^\gamma y^\beta(t)=0, t \in ]0,1[
subject to the following boundary conditions
y(0)=1,~y(1)=0; y(0)=1,~y'(1)=y(1),
where \gamma, \beta and \sigma are real numbers, \gamma<-2, \beta>1. We propsoed to solve the above BVPs with the aid of Haar wavelet coupled with q...

The existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. The fourth-order radial equation is non-self adjoint and has no exact solutions. Also, it admits multiple solutions. Furthermore, solutions depend on the size of the parameter. We show that s...

We propose exact finite difference scheme (EFD) for Generalized Burgers Fisher (GBF) Equation $w_xx=w_t+αw^θ w_x−βw(1−w^θ), 0≤x≤1, t≥0, using solitary wave solution. Moreover a non-standard finite difference (NSFD) scheme is also proposed. The proposed EFD and NSFD scheme works for all θ∈N. This scheme preserves the positivity and boundedness prope...

In this paper we have considered generalized Emden-Fowler equation, \begin{equation*} y''(t)+\sigma t^\gamma y^\beta(t)=0, ~~~~~~~~t \in ]0,1[ \end{equation*} subject to the following boundary conditions \begin{equation*} y(0)=1,~y(1)=0;~~\&~~y(0)=1,~y'(1)=y(1), \end{equation*} where $\gamma,\beta$ and $\sigma$ are real numbers, $\gamma<-2$, $\beta...

The existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. The fourth-order radial equation is non-self adjoint and has no exact solutions. Also, it admits multiple solutions. Furthermore, solutions depend on the size of the parameter. We show that s...

In this paper we derive $7^{th}$ order convergent integration formula in time which is weakly $L$-stable. To derive the method we use, Newton Cotes formula, fifth-order Hermite interpolation polynomial approximation (osculatory interpolation) and sixth-order explicit backward Taylor's polynomial approximation. The vector form of this formula is use...

Purpose
This paper aims to apply an iterative numerical method to find the numerical solution of the nonlinear non-self-adjoint singular boundary value problems that arises in the theory of epitaxial growth.
Design/methodology/approach
The proposed problem has multiple solutions and it is singular too; so not every technique can capture all the so...

We propose a new class of SBVPs which deals with exothermic reactions. We also propose four computationally stable methods to solve singular nonlinear BVPs by using Hermite wavelet collocation which are coupled with Newton's quasilinearization and Newton-Raphson method. We compare the results obtained with Hermite Wavelets with Haar wavelet colloca...

Getting solution near singular point of any non-linear BVP is always tough because solution blows up near singularity. In this article our goal is to construct a general method based on orthogonal polynomial and then use different orthogonal polynomials as particular wavelets. To show importance and accuracy of our method we have solved non-linear...

In this paper, we consider a non-self-adjoint, singular, nonlinear fourth order boundary value problem which arises in the theory of epitaxial growth. It is possible to reduce the fourth order equation to a singular boundary value problem of second order given by
w''-(1/r)w'=(w^2/2r^2)+(1/2)\lambda r^2
The problem depends on the parameter λ and a...

Consider the class of four point nonlinear BVPs
−𝑤″(𝑥)=𝑓(𝑥,𝑤,𝑤′), 𝑥∈𝐼,
𝑤′(0)=0, 𝑤(1)=𝛿_1𝑤(𝜂_1)+𝛿_2𝑤(𝜂_2),
where 𝑓∈(𝐼×ℝ×ℝ,ℝ) is continuous, 𝐼=[0,1], 𝜂_1,𝜂_2∈(0,1) such that 𝜂_1≤𝜂_2 and 𝛿_1,𝛿_2≥0. In this paper, we demonstrate an iterative technique. The iterative scheme is deduced by using quasilinearization. Then we consider upper-lower solutions i...

In the original article, the analytical solution of Problem 9 is published incorrectly.

In this article, we propose a novel modification to Quasi-Newton method, which is now a days popularly known as variation iteration method (VIM) and use it to solve the following class of nonlinear singular differential equations which arises in chemistry
\[-y''(x)-\frac{\alpha}{x}y'(x)=f(x,y),~x\in(0,1),\]
where $\alpha\geq1$, subject to certain t...

Computing solutions of singular differential equations has always been a challenge as near the point of singularity it is extremely difficult to capture the solution. In this research article, Haar wavelet coupled with quasilinearization approach (HWQA) is proposed for computing numerical solution of non-linear SBVPs popularly also referred as Lane...

In this paper we consider the following class of four point boundary value
problems
—y"(x) = f (x, y), 0 < x < 1,
y'(0) = 0, y(1) = \delta_1y(\eta_1) + \delta_2y(\eta_2),
where \delta_1, \delta_2 \geq 0 , 0<\eta_1, \eta_2 < 1, and f (x, y), is continuous in one sided Lipschitz in y. We propose a monotone iterative scheme and show that under some
su...

In this paper, we analyse Mickens’ type non-standard finite difference schemes (NSFD) and establish their convergence. We then apply these schemes on Lane Emden equations. The numerical results thus obtained are compared with existing analytical solutions or with solutions computed with standard finite difference (FD) schemes. NSFD and FD solutions...

A monotone iterative method is proposed to solve nonlinear discrete boundary value problems with the support of upper and lower solutions. We establish some new existence results. Under some sufficient conditions, we establish maximum principle for linear discrete boundary value problem, which relies on Green's function and its constant sign. We th...

In this work, we propose an effective numerical technique for a class of Lane-Emden equation which arises in chemistry and other branches. This technique is the combination of variational iteration and homotopy perturbation. It produces approximate solution in the form of a series, which is very handy from computational point of view. Accuracy of t...

This article deals with a computational iterative technique for the following second order three point boundary value problem
y''(t) + f(t, y, y' ) = 0, 0 < t < 1,
y(0) = 0, y(1) = δy(η),
where f(I × R, R), I = [0, 1], 0 < η < 1, δ > 0. We consider simple iterative scheme and develop a monotone iterative technique. Some examples are constructed...

We consider the following class of nonlinear three point singular boundary value problems (SBVPs) -y''(x)-(2/x)y'(x) = f(x, y), 0 < x < 1, y'(0) = 0, y(1) = δy(η), where δ> 0 and 0 <η< 1. We establish some new maximum principles. Further using these maximum principles and monotone iterative technique in the presence of upper and lower solution we p...

This article deals with a class of three-point nonlinear boundary-value problems
(BVPs) with Neumann type boundary conditions which arises in bridge design.
The source term (nonlinear term) depends on the derivative of the solution,
it is also Lipschitz continuous. We use monotone iterative technique in the
presence of upper and lower solutions for...

AbstractWe derive fifth order convergent method in time for the initial value problem When applied to test equation y?=-?y, ?>0 it gives yn+1=?(z)yn where ?(z) does not satisfy the condition for A-stability but ?(z)?0 as z?∞. To develop this method we use a higher order average approximation which is based on osculatory cubic polynomial interpolati...

We prove maximum and anti-maximum principle for the following differential inequalities, (Formula Presented.) where (Formula Presented.) and (Formula Presented.) and use it to examine the existence of solutions of the following class of nonlinear three point singular boundary value problems (SBVPs) (Formula Presented.) We use monotone iterative tec...

In this work we deal with a nonlinear three point singular boundary value problems (SBVPs), when the nonlinearity depends upon derivative. We establish the maximum principles for linear model. Prove some new inequalities based on Bessel and modified Bessel functions. Finally by using the Monotone Iterative Technique, we obtain some new existence re...

In this article we consider the following class of three point boundary value problem
and use monotone iterative technique to derive some sufficient conditions of existence. Examples are included to illustrate the effectiveness of the proposed results. We consider both well ordered and reverse ordered upper and lower solutions.

In this paper we apply the Du Fort–Frankel finite difference scheme on Burgers equation and solve three test problems. We calculate the numerical solutions using Mathematica 7.0 for different values of viscosity. We have considered smallest value of viscosity as 10−4 and observe that the numerical solutions are in good agreement with the exact solu...

We consider the following class of three point boundary value problem y '' (t)+f(t,y)=0, 0<t<1, y ' (0)=0, y(1)=δy(η) where δ>0, 0<η<1, the source term f(t,y) is Lipschitz and continuous. We use monotone iterative technique in the presence of upper and lower solutions for both well-order and reverse order cases. Under some sufficient conditions, we...

In this paper we consider the class of nonlinear singular differential equations of the type-p(x)y′(x)′+q(x)fx,y(x),p(x)y′(x)=0,0<x<1,subject to the boundary conditionslimx→0p(x)y′(x)=0,limx→1p(x)y′(x)=0.Conditions on p(x)p(x) and q(x)q(x) are imposed so that x=0x=0 is a regular singular point. An approximation scheme which is iterative in nature i...

In this paper we consider a class of nonlinear singular boundary value problems of the type
where 1≤α≤β. When β=α and the nonlinear source function is f(x,y) or linear this problem arises in physiology (Adam and Maggelakis in Math. Biosci. 97:121–136, 1989; Anderson and Arthurs in Bull. Math. Biol. 47(1):145–153, 1985; Duggan and Goodman in Bull....

Monotone iterative technique is employed for studying the existence of solutions to the second-order nonlinear singular boundary value problem -p (x) y ' (x) ' +p(x)fx , y (x) , p (x) y ' (x)=0 for 0<x<1 and y ' (0)=y ' (1)=0. Here, p(0)=0 and xp ' (x)/p(x) is analytic at x=0. The source function f(x,y,py ' ) is Lipschitz in py ' and one sided Lips...

In this paper we develop an unconditionally stable third order time integration formula for the diffusion equation with Neumann boundary condition. We use a suitable arithmetic average approximation and explicit backward Euler formula and then develop a third order L-stable Simpson’s 3/8 type formula. We also observe that the arithmetic average app...

We consider the following class of nonlinear singular differential equation - ( p ( x ) y ′ ( x ) ) ′ + q ( x ) f ( x , y ( x ) , p ( x ) y ′ ( x ) ) = 0, 0 < x < 1 subject to the Neumann boundary condition y ′ ( 0 ) = y ′ ( 1 ) = 0 . Conditions on p ( x ) and q ( x ) ensure that x = 0 is a singular point of limit circle type. A simple approximatio...

In this paper we establish existence of solutions of singular boundary value problem −(p(x)y
′(x))′=q(x)f(x,y,py′) for 0<x≤b and
limx®0+p(x)y¢(x)=0\lim_{x\rightarrow0^{+}}p(x)y^{\prime}(x)=0
, α
1
y(b)+β
1
p(b)y
′(b)=γ
1 with p(0)=0 and q(x) is allowed to have integrable discontinuity at x=0. So the problem may be doubly singular. Here we consider...

## Questions

Question (1)

How to differentiate between exact Solution and analytical solution?