Solutions of the system of differential equations by differential transform method
TL;DR: Three-dimensional differential transform method has been introduced and fundamental theorems have been defined for the first time and exact solutions of linear and non-linear systems of partial differential equations have been investigated.
About: This article is published in Applied Mathematics and Computation.The article was published on 2004-01-01. It has received 383 citations till now. The article focuses on the topics: Numerical partial differential equations & Laplace transform applied to differential equations.
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TL;DR: To the best of our knowledge, there is only one application of mathematical modelling to face recognition as mentioned in this paper, and it is a face recognition problem that scarcely clamoured for attention before the computer age but, having surfaced, has attracted the attention of some fine minds.
Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.
3,015 citations
TL;DR: In this article, the effects of variable surface heat flux and first-order chemical reaction on MHD flow and radiation heat transfer of nanofluids against a flat plate in porous medium were investigated.
317 citations
TL;DR: In this paper, the effect of the squeeze number, nanofluid volume fraction, Hartmann number and heat source parameter on flow and heat transfer was investigated, and the results showed that skin friction coefficient increases with increase of the Nusselt number and Hartmann numbers but it decreases with an increase in the volume fraction.
311 citations
TL;DR: In this paper, Least Square and Galerkin methods are used to solve the problem of laminar nanofluid flow in a semi-porous channel in the presence of transverse magnetic field.
254 citations
TL;DR: A new generalization of the one-dimensional differential transform method that will extend the application of the method to differential equations of fractional order is proposed, based on generalized Taylor’s formula and Caputo fractional derivative.
216 citations
References
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Book•
01 Jan 1974
TL;DR: In this paper, a general overview of the nonlinear theory of water wave dynamics is presented, including the Wave Equation, the Wave Hierarchies, and the Variational Method of Wave Dispersion.
Abstract: Introduction and General Outline. HYPERBOLIC WAVES. Waves and First Order Equations. Specific Problems. Burger's Equation. Hyperbolic Systems. Gas Dynamics. The Wave Equation. Shock Dynamics. The Propagation of Weak Shocks. Wave Hierarchies. DISPERSIVE WAVES. Linear Dispersive Waves. Wave Patterns. Water Waves. Nonlinear Dispersion and the Variational Method. Group Velocities, Instability, and Higher Order Dispersion. Applications of the Nonlinear Theory. Exact Solutions: Interacting Solitary Waves. References. Index.
8,808 citations
TL;DR: To the best of our knowledge, there is only one application of mathematical modelling to face recognition as mentioned in this paper, and it is a face recognition problem that scarcely clamoured for attention before the computer age but, having surfaced, has attracted the attention of some fine minds.
Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.
3,015 citations
Book•
19 Aug 1997TL;DR: The Third edition of the Third Edition of as discussed by the authors is the most complete and complete version of this work. But it does not cover the first-order nonlinear Equations and their applications.
Abstract: Preface to the Third Edition.- Preface.- Linear Partial Differential Equations.- Nonlinear Model Equations and Variational Principles.- First-Order, Quasi-Linear Equations and Method of Characteristics.- First-Order Nonlinear Equations and Their Applications.- Conservation Laws and Shock Waves.- Kinematic Waves and Real-World Nonlinear Problems.- Nonlinear Dispersive Waves and Whitham's Equations.- Nonlinear Diffusion-Reaction Phenomena.- Solitons and the Inverse Scattering Transform.- The Nonlinear Schroedinger Equation and Solitary Waves.- Nonlinear Klein--Gordon and Sine-Gordon Equations.- Asymptotic Methods and Nonlinear Evolution Equations.- Tables of Integral Transforms.- Answers and Hints to Selected Exercises.- Bibliography.- Index.
744 citations
"Solutions of the system of differen..." refers methods in this paper
...As mentioned in [4], solution of this problem was done before by [1] but after cumbersome work....
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TL;DR: Using two-dimensional differential transform to solve Partial Differential Equations (PDE) is proposed in this study and three PDE problems with constant and variable coefficients are solved.
334 citations
Additional excerpts
...The basic definitions and fundamental Theorems 1–7 and 8–14 of the twodimensional transform are defined in [6,8] respectively and given as follows: W ðk; hÞ 1⁄4 1 k!h! okþhwðx; yÞ oxkoyh ð0;0Þ ; ð1Þ where wðx; yÞ is the original function and W ðk; hÞ is the transformed function....
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Book•
01 Jan 1994
TL;DR: In this article, the authors present an approach to the problem of finding a solution to the first order differential equation in a set of linear equations with respect to the velocity of the wave.
Abstract: Preface. 1. Partial Differential Equations. 1.1 Partial Differential Equations. 1.1.1 PDEs and Solutions. 1.1.2 Classification. 1.1.3 Linear vs. Nonlinear. 1.1.4 Linear Equations. 1.2 Conservation Laws. 1.2.1 One Dimension. 1.2.2 Higher Dimensions. 1.3 Constitutive Relations. 1.4 Initial and Boundary Value Problems. 1.5 Waves. 1.5.1 Traveling Waves. 1.5.2 Plane Waves. 1.5.3 Plane Waves and Transforms. 1.5.4 Nonlinear Dispersion. 2. First-Order Equations and Characteristics. 2.1 Linear First-Order Equations. 2.1.1 Advection Equation. 2.1.2 Variable Coefficients. 2.2 Nonlinear Equations. 2.3 Quasi-linear Equations. 2.3.1 The general solution. 2.4 Propagation of Singularities. 2.5 General First-Order Equation. 2.5.1 Complete Integral. 2.6 Uniqueness Result. 2.7 Models in Biology. 2.7.1 Age-Structure. 2.7.2 Structured predator-prey model. 2.7.3 Chemotherapy. 2.7.4 Mass structure. 2.7.5 Size-dependent predation. 3. Weak Solutions To Hyperbolic Equations. 3.1 Discontinuous Solutions. 3.2 Jump Conditions. 3.2.1 Rarefaction Waves. 3.2.2 Shock Propagation. 3.3 Shock Formation. 3.4 Applications. 3.4.1 Traffic Flow. 3.4.2 Plug Flow Chemical Reactors. 3.5 Weak Solutions: A Formal Approach. 3.6 Asymptotic Behavior of Shocks. 3.6.1 Equal-Area Principle. 3.6.2 Shock Fitting. 3.6.3 Asymptotic Behavior. 4. Hyperbolic Systems. 4.1 Shallow Water Waves Gas Dynamics. 4.1.1 Shallow Water Waves. 4.1.2 Small-Amplitude Approximation. 4.1.3 Gas Dynamics. 4.2 Hyperbolic Systems and Characteristics. 4.2.1 Classification. 4.3 The Riemann Method. 4.3.1 Jump Conditions for Systems. 4.3.2 Breaking Dam Problem. 4.3.3 Receding Wall Problem. 4.3.4 Formation of a Bore. 4.3.5 Gas Dynamics. 4.4 Hodographs and Wavefronts. 4.4.1 Hodograph Transformation. 4.4.2 Wavefront Expansions. 4.5 Weakly Nonlinear Approximations. 4.5.1 Derivation of Burgers' Equation. 5. Diffusion Processes. 5.1 Diffusion and Random Motion. 5.2 Similarity Methods. 5.3 Nonlinear Diffusion Models. 5.4 Reaction-Diffusion Fisher's Equation. 5.4.1 Traveling Wave Solutions. 5.4.2 Perturbation Solution. 5.4.3 Stability of Traveling Waves. 5.4.4 Nagumo's Equation. 5.5 Advection-Diffusion Burgers' Equation. 5.5.1 Traveling Wave Solution. 5.5.2 Initial Value Problem. 5.6 Asymptotic Solution to Burgers' Equation. 5.6.1 Evolution of a Point Source. 6. Reaction-Diffusion Systems. 6.1 Reaction-Diffusion Models. 6.1.1 Predator-Prey Model. 6.1.2 Combustion. 6.1.3 Chemotaxis. 6.2 Traveling Wave Solutions. 6.2.1 Model for the Spread of a Disease. 6.2.2 Contaminant transport in groundwater. 6.3 Existence of Solutions. 6.3.1 Fixed-Point Iteration. 6.3.2 Semi-Linear Equations. 6.3.3 Normed Linear Spaces. 6.3.4 General Existence Theorem. 6.4 Maximum Principles. 6.4.1 Maximum Principles. 6.4.2 Comparison Theorems. 6.5 Energy Estimates and Asymptotic Behavior. 6.5.1 Calculus Inequalities. 6.5.2 Energy Estimates. 6.5.3 Invariant Sets. 6.6 Pattern Formation. 7. Equilibrium Models. 7.1 Elliptic Models. 7.2 Theoretical Results. 7.2.1 Maximum Principle. 7.2.2 Existence Theorem. 7.3 Eigenvalue Problems. 7.3.1 Linear Eigenvalue Problems. 7.3.2 Nonlinear Eigenvalue Problems. 7.4 Stability and Bifurcation. 7.4.1 Ordinary Differential Equations. 7.4.2 Partial Differential Equations. References. Index.
328 citations