Mathematical models are used to convert real-life problems using mathematical concepts and language. These models are governed by differential equations whose solutions make it easy to understand real-life problems and can be applied to engineering and science disciplines. This book presents numerical methods for solving various mathematical models.
This book offers real-life applications, includes research problems on numerical treatment, and shows how to develop the numerical methods for solving problems. The book also covers theory and applications in engineering and science.
Engineers, mathematicians, scientists, and researchers working on real-life mathematical problems will find this book useful.
| Preface |
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xi | |
| About the Author |
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xv | |
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Chapter 1 Stability and Convergence Analysis of Numerical Scheme for the Generalized Fractional Diffusion-Reaction Equation |
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1 | (16) |
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2 | (1) |
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1.2 Fractional Derivatives Review |
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3 | (2) |
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1.3 Existence and Uniqueness Via Banach Fixed Theorem |
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5 | (2) |
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1.4 Numerical Scheme of the Fractional Diffusion Reaction Equation |
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7 | (1) |
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1.5 Stability Analysis of the Numerical Approximation |
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8 | (2) |
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1.6 Convergence Analysis of the Numerical Approximation |
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10 | (2) |
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1.7 The Graphics with the Numerical Scheme |
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12 | (2) |
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14 | (1) |
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14 | (3) |
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Chapter 2 Studying on the Complex and Mixed Dark-Bright Travelling Wave Solutions of the Generalized KP-BBM Equation |
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17 | (22) |
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17 | (2) |
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19 | (1) |
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2.3 Applications of SGEM and Mathematical Analysis |
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20 | (2) |
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2.3.1 Investigation of Generalized KP-BBM Equation |
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20 | (2) |
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22 | (13) |
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35 | (4) |
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Chapter 3 Abundant Computational and Numerical Solutions of the Fractional Quantum Version of the Relativistic Energy-Momentum Relation |
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39 | (48) |
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40 | (3) |
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3.2 Analytical Explicit Wave Solutions |
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43 | (28) |
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3.2.1 Extended exp (--φ(Ξ)) Expansion Method |
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43 | (6) |
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3.2.2 Extended Fan Expansion Method |
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49 | (13) |
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3.2.3 Extended (G'/G) Expansion Method |
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62 | (2) |
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3.2.4 Improved F-expansion Method |
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64 | (3) |
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3.2.5 Modified Khater Method |
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67 | (4) |
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71 | (1) |
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72 | (6) |
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3.4.1 Semi-Analytical Solutions |
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72 | (1) |
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3.4.2 Numerical Solutions |
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73 | (1) |
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74 | (2) |
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3.4.2.2 Quantic--B--spline |
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76 | (1) |
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77 | (1) |
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3.5 Figures Representation |
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78 | (3) |
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81 | (1) |
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81 | (6) |
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Chapter 4 Applications of Conserved Schemes for Solving Ultra-Relativistic Euler Equations |
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87 | (22) |
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87 | (2) |
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89 | (3) |
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4.2.1 The (p, u) Subsystem |
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90 | (2) |
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4.3 The Numerical Schemes |
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92 | (3) |
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92 | (1) |
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4.3.2 The Structure of Numerical Solutions |
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93 | (2) |
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95 | (9) |
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104 | (1) |
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105 | (4) |
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Chapter 5 Notorious Boundary Value Problems: Singularly Perturbed Differential Equations and Their Numerical Treatment |
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109 | (26) |
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110 | (2) |
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112 | (16) |
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5.2.1 A Priori Refined Meshes |
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113 | (1) |
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5.2.1.1 Bakhvalov-Type Meshes |
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113 | (2) |
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5.2.1.2 Shishkin-Type Meshes |
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115 | (1) |
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5.2.1.3 Comparison Between Bakhvalov Mesh and Shishkin Mesh |
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116 | (3) |
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5.2.2 A Posteriori Refined Meshes |
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119 | (3) |
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5.2.3 Error Estimates and the Construction of A Monitor Function |
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122 | (1) |
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5.2.3.1 Constructing a Monitor Function from a Priori Error Estimates |
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122 | (1) |
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5.2.3.2 Constructing a Monitor Function from a Posteriori Error Estimates |
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123 | (2) |
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5.2.4 Numerical Experiments for Mesh Adaptation on a Test Problem |
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125 | (3) |
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128 | (4) |
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132 | (1) |
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132 | (3) |
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Chapter 6 Review on Non-Standard Finite Difference (NSFD) Schemes for Solving Linear and Non-linear Differential Equations |
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135 | (20) |
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135 | (7) |
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6.2 Non-standard Finite Difference (NSFD) Schemes |
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142 | (7) |
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6.2.1 Comparison between Standard and Non-Standard Finite Difference Methods |
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143 | (1) |
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6.2.2 Applications of NSFD scheme |
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143 | (1) |
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6.2.2.1 Applications to Modelled ODEs |
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144 | (3) |
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6.2.2.2 Applications to Modelled PDEs |
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147 | (1) |
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6.2.2.3 Applications to Modelled Fractional Differential Equations |
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148 | (1) |
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6.3 Conclusions and Scope |
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149 | (2) |
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151 | (4) |
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Chapter 7 Solutions for Nonlinear Fractional Diffusion Equations with Reaction Terms |
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155 | (32) |
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155 | (2) |
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7.2 Reaction Diffusion Problem |
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157 | (8) |
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157 | (5) |
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162 | (3) |
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165 | (16) |
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166 | (7) |
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173 | (8) |
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181 | (2) |
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183 | (1) |
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183 | (4) |
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Chapter 8 Convergence of Some High-Order Iterative Methods with Applications to Differential Equations |
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187 | (18) |
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187 | (2) |
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8.2 Local Convergence Analysis |
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189 | (9) |
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198 | (3) |
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201 | (1) |
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202 | (1) |
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202 | (3) |
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Chapter 9 Fractional Derivative Operator on Quarantine and Isolation Principle for COVID-19 |
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205 | (22) |
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205 | (5) |
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9.2 Mathematical Analysis of the Dynamics |
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210 | (6) |
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9.2.1 Uniqueness and Continuous Dependence of the Solution |
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213 | (1) |
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9.2.2 Equilibrium for the Dynamics |
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214 | (2) |
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9.3 Derivation of the Numerical Method |
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216 | (3) |
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9.4 Numerical Results and Discussions |
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219 | (3) |
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222 | (1) |
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223 | (1) |
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223 | (4) |
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Chapter 10 Superabundant Explicit Wave and Numerical Solutions of the Fractional Isotropic Extension Model of the KdV Model |
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227 | (52) |
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228 | (1) |
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10.2 Analytical Explicit Wave Solutions |
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229 | (23) |
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10.2.1 Exp(--φ(Ξ))-Expansion Method |
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229 | (4) |
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10.2.2 Extended Fan-Expansion Method |
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233 | (7) |
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10.2.3 Extended (G'/G)-Expansion Method |
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240 | (3) |
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10.2.4 Extended Simplest Equation Method |
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243 | (2) |
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10.2.5 Extended Tanh(Ξ)-Expansion Method |
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245 | (2) |
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10.2.6 Modified Khater Method |
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247 | (5) |
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252 | (1) |
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253 | (5) |
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10.4.1 Semi-Analytical Solutions |
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253 | (2) |
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10.4.2 Numerical Solutions |
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255 | (1) |
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255 | (1) |
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10.4.2.2 Quantic B-Spline |
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256 | (2) |
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258 | (1) |
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10.5 Figures and Tables Representation |
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258 | (17) |
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275 | (1) |
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275 | (4) |
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Chapter 11 A Modified Computational Scheme and Convergence Analysis for Fractional Order Hepatitis E Virus Model |
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279 | (34) |
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279 | (2) |
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11.2 Elemental Definitions and Formulae |
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281 | (1) |
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11.3 Mathematical Description Of Hev Model |
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282 | (3) |
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11.4 q-HASTM: Basic Methodology |
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285 | (4) |
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11.5 Uniqueness and Convergence Analysis for q-HASTM |
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289 | (2) |
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11.6 q-HASTM Solution for the Fractional Hev Model |
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291 | (11) |
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11.7 Numerical Simulations |
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302 | (7) |
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11.8 Concluding Remarks and Observations |
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309 | (1) |
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310 | (3) |
| Index |
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313 | |
Harendra Singh is an Assistant Professor in the Department of Mathematics, Post-Graduate College, Ghazipur-233001, Uttar Pradesh, India. He did his Master of Science (M.Sc.) in Mathematics from Banaras Hindu University, Varanasi and Ph.D. in Mathematics from Indian Institute of Technology (BHU), Varanasi, India. He has qualified GATE, JRF and NBHM in Mathematics. He is also awarded by post-doctoral fellowship (PDF) in Mathematics from National Institute of Science Education and Research (NISER) Bhubaneswar Odisha, India. He primarily teaches the subjects like real and complex analysis, functional analysis, Abstract Algebra and measure theory in post-graduate level course in mathematics. His area of interest is Mathematical Modelling, Fractional Differential Equations, Integral Equations, Calculus of Variations, Analytical and Numerical Methods. His works have been published in Applied Numerical Mathematics, Applied Mathematics and Computations, Applied Mathematical Modelling, Chaos Solitions & Fractals, Numerical Methods for Partial Differential Equations, Physica A, Astrophysics and Space Science, Electronic Journal of Differential Equations, Few Body-system and several other peer-reviewed international journals. He has edited a book published by CRC press Taylor and Francis. His 32 research papers have been published in various Journals of repute with h-index of 11. His sole author papers have been published in Applied Mathematics and Computations, Chaos Solitions & Fractals, Astrophysics and Space Science and other peer-reviewed international journals. He has attained a number of National and International Conferences and presented several research papers. He has also attended Short Terms Programs and Workshops. He is reviewer of various Journals.
Jagdev Singh is an Professor in the Department of Mathematics, JECRC University, Jaipur-303905, Rajasthan, India. He did his Master of Science (M.Sc.) in Mathematics and Ph.D. in Mathematics from University of Rajasthan, India. He primarily teaches the subjects like mathematical modeling, real analysis, functional analysis, integral equations and special functions in post-graduate level course in mathematics. His area of interest is Integral Transforms, Special Functions, Fractional Calculus, Mathematical Modelling, Mathematical Biology, Fluid Dynamics, Applied Functional Analysis, Nonlinear Dynamics, Analytical and Numerical Methods. He has published three books: Advance Engineering Mathematics (2007), Engineering Mathematics-I (2008), Engineering Mathematics-II (2013). His works have been published in the Nonlinear Dynamics, Chaos Solitions & Fractals, Physica A, Journal of Computational and Nonlinear Dynamics, Applied Mathematical Modelling, Entropy, Advances in Nonlinear Analysis, Romanian Reports in Physics, Applied Mathematics and Computation, Chaos and several other peer-reviewed international journals. His 120 research papers have been published in various Journals of repute with h-index of 30. He has attained a number of National and International Conferences and presented several research papers. He has also attended Summer Courses, Short Terms Programs and Workshops. He is member of Editorial Board of various Journals of Mathematics. He is reviewer of various Journals.
Sunil Dutt Purohit is an Associate professor of Mathematics at Rajasthan Technical University, Kota, India. He did his Master of Science (M.Sc.) in Mathematics and Ph.D. in Mathematics from Jai Narayan Vyas University, Jodhpur, India. He was awarded University Gold Medal for being topper in M.Sc. Mathematics and awarded Junior Research Fellowship and Senior Research Fellow of Council of Scientific and Industrial Research. He primarily teaches the subjects like integral transforms, complex analysis, numerical analysis and optimization techniques in graduate and post-graduate level course in engineering mathematics. His research interest includes special functions, basic hypergeometric series, fractional calculus, geometric function theory, mathematical analysis and modelling. He's credited more than 150 research articles and four books so far. He has delivered talks at foreign and national institutions. He has also organized a number of academic events. He is a Life Member of Indian Mathematical Society (IMS), Indian Science Congress Association (ISCA), Indian Academy of Mathematics (IAM) and Society for Special Functions and their Applications, Soft Computing Research Society, India (SCRS) and International Association of Engineers (IAENG). Presently, he is general-secretary of the Rajasthan Ganita Parishad. He has also contributed in designing and redesigning of syllabus of engineering mathematics B. Tech. course work.
Devendra Kumar is an Assistant Professor in the Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India. He did his Master of Science (M.Sc.) in Mathematics and Ph.D. in Mathematics from University of Rajasthan, India. He primarily teaches the subjects like real and complex analysis, functional analysis, integral equations and special functions in post-graduate level course in mathematics. His area of interest is Integral Transforms, Special Functions, Fractional Calculus, Mathematical Modelling, Applied Functional Analysis, Nonlinear Dynamics, Analytical and Numerical Methods. He has published two books: Engineering Mathematics-I (2008), Engineering Mathematics-II (2013). His works have been published in the Nonlinear Dynamics, Chaos Solitions & Fractals, Physica A, Journal of Computational and Nonlinear Dynamics, Applied Mathematical Modelling, Entropy, Advances in Nonlinear Analysis, Romanian Reports in Physics, Applied Mathematics and Computation, Chaos and several other peer-reviewed international journals. His 130 research papers have been published in various Journals of repute with h-index of 30. He has attained a number of National and International Conferences and presented several research papers. He has also attended Summer Courses, Short Terms Programs and Workshops. He is member of Editorial Board of various Journals of Mathematics. He is reviewer of various Journals.
Rohkem infot Taylor & Francis e-raamatute kohta